Yes, from a natural storm.
For both a conjured and a natural storm, there isn't a provision in the rules for passing concentration on a spell (any spell) from one caster to another, and the only thing one "controls" with the spell is the lightning called down to do damage.
Use the game mechanic of Concentration to determine what 'control' of the storm means.
Concentration (Basic Rules p. 79-80)
Some spells require you to maintain concentration in order to keep their magic active. If you lose concentration, such a spell ends.
What is controlled by call lightning? This is a third level spell, while control weather is an 8th level spell. (More on that below).
The two cases of call lightning: caster-summoned and natural storms
Call Lightning requires concentration for up to 10 minutes after a storm cloud is summoned. A spell caster controls a storm that was formed by this spell as long as he concentrates on it. When concentration ends, the storm ends.
For a natural storm outdoors (stormy weather) there is no obstacle to more than one spell caster calling down lightning from it (details below). Depending upon the situation, either the Contest rules or the Combining Magical Effects rules may need to be used. (see below).
What's the significant difference?
The distinction between Case 1 and Case 2 is based upon the storm's formation. In both cases, what is controlled by the spell is the lightning called down from the storm to strike targets.
A storm that was formed by the spell goes away when concentration
ends.
When a spell caster stops concentrating on the natural storm, the
storm remains.
The short answer for Case 1 is that the rules for concentration render moot any contest for control of the conjured storm.
The short answer for Case 2 is that the spell is irrelevant to the storm's existence. When a spell caster's concentration ends, the storm is still there. What has changed is that the spell caster no longer calls down lightning from the storm. To actually control the natural storm, a spell of greater power (control weather, 8th level spell) fits the requirements better.
Discussion
Case 1. There isn't a natural storm
From the spell description, only the spell caster's concentrating on the conjured storm allows the caster to call down lightning from it. No provision is made for passing control of the storm to someone else, nor of passing concentration on a spell to another spell caster. (Note: I did not find any spell in the books where one spell caster can pass concentration an a spell to another spell caster).
What is controlled? The spell caster controls where the lightning strikes. By looking at this spell as a spell requiring concentration the issue of control doesn't arise in Case 1.
Does the conjured storm cloud move with the spell caster or not?
No, per Mike Mearls.
4 Dec 2015 @TheShieldComics
@mikemearls When call lightning is cast, does the storm follow the caster when he moves, staying centered directly on him?
11:13 AM - 4 Dec 2015 @mikemearls
@TheShieldComics the storm does not move - > stays at its starting point
The point that another spell caster can't control someone else's conjured storm cloud is consistent with the absence of any rule to pass concentration on a spell from one caster to another. Control in case 1 isn't even contestable.
Bottom Line: this is a Concentration spell. At best, a second spell caster can try to break the first spell caster's concentration and end the spell by inflicting damage on the original spell caster.
Case 2. There's an actual storm, and two spell casters are outdoors.
Why is "control" mentioned in case 2? (Good question!) Case 2 mentions getting control of an existing {natural} storm cloud, rather than conjuring a storm that wasn't already there. This storm is used to call down lightning.
The original spell caster did not conjure an artificial storm cloud because one already exists. The spell caster calls down lightning from a storm to strike targets within range. The caster concentrates on the spell, but did not have to create a storm. The text describes this condition as "control."
What is the caster controlling? All that the text specifies is control of where the lightning strikes. Control in terms of how fast, how far, and where the storm can be moved isn't specified in the text.
There is a lot not specified in this spell description.
How big is the natural storm/storm cloud? Unspecified.
You could infer that "control" refers to the whole storm and that wherever the caster moves under the storm she can strike a target within range. Moving while under the storm would give the caster great flexibility. That isn't specified in the text. (But control weather is an 8th level spell, and call lightning is a 3rd level spell).
How hard is it to control a storm?
Compare this 3rd level spell, call lightning, to control weather, an 8th level spell. The latter spell lets you create a storm within five miles of you, and includes text that specifies what is controlled. The only thing specified as under the spell caster's control in call lightning is where the lightning hits and how much harder it hits than the conjured storm. That amounts to the same level of control as in Case 1, with a damage bonus.
As long as you concentrate on this spell, you control where the lightning strikes.
Can two spell casters concentrate on the same natural storm?
That is not specified, but it is also not prohibited.
Neither spell caster is using magic on the order of control weather (an 8th level spell). With that level of power as a reference point, the spell caster controlling the entire storm seems to be beyond the power of a third level spell.
What if both casters are within range of each other and both are trying to do the same thing at once to the same storm?
In that specific case, the Contest rules should cover this. A successful contest by the challenger moves local control of the storm to him. A tie or a loss by the challenger leaves the original spell caster in control of the lightning strikes as long as her concentration is not broken. That is consistent with a rule as written, but isn't specific to the call lightning rule text.
Contests (p. 58 Basic Rules)
Sometimes one character’s or monster’s efforts are directly opposed to
another’s. This can occur when both of them are trying to do the same
thing and only one can succeed ... In situations like these, the
outcome is determined by a special form of ability check, called a
contest. {snip} The participant with the higher check
total wins the contest. That character or monster either succeeds at
the action or prevents the other one from succeeding. If the contest
results in a tie, the situation remains the same as it was before the
contest. Thus, one contestant might win the contest by default.
What if a challenger casts a fifth level call lightning spell while another spell caster is concentrating on the storm using a third level slot for the spell?
Either they are both trying to do the same thing at once, and a Contest is the way to go, or you are dealing with Combining Magical Effects.
(p. 81 Basic Rules) The effects of different spells add together while the durations of those spells overlap. The effects of the same spell cast multiple times don’t combine, however. Instead, the most potent effect—such as the highest bonus—from those castings applies while their durations overlap. For example, if two clerics cast bless on the same target, that character gains the spell’s benefit only once; he or she doesn’t get to roll two bonus dice.
Those are the tools available in the rules to address that situation.
What is "Control" for this spell?
the spell gives you control over the existing storm instead of creating a new one.
What did "creating a new one" allow the spell caster to control? Getting lightning out of the storm and directing where the lightning hits. The spell caster's control is thus limited to where the lightning hits in both kinds of storm. By understanding the spell in terms of a game mechanic, Concentration, and comparing the level of magic needed to control an entire storm (an eighth level spell), control's ambiguous meaning is resolved.
What if two spell casters both want to call lightning out of the same storm, and they are a few miles apart?
Neither caster is controlling weather. Their spell effects won't trigger the overlapping spell effects rule. They both call down lightning without interference as long as they concentrate on the spell.
Another way this could happen in the outdoors:
Two opposed spell casters send lightning bolts at each other (in range) while concentrating on different bits of the storm that don't overlap. The first one to fail the Constitution save due to damage loses Concentration on the spell -- at that point, control of the storm is moot. Using that illustration, the point on limited control becomes stronger.
Summary
If the call lightning spell conjures its own storm cloud, only the original spell caster's concentration keeps the storm cloud available from which to call down lightning. When the spell caster's concentration ends, the artificial storm cloud ends.
If a storm is already in existence outdoors, the spell caster concentrates on it. Control is limited to getting lighting to come down within range. When spell concentration ends, the natural storm is still there.
One last consideration
Outdoors, even without a storm, in ten minutes you could call down 100 (10 turns in a minute for ten minutes) lightning strikes on a building or a number of buildings, doing 300d10 damage in a 60'radius if nobody broke your concentration. You would do 400d10 during a natural storm. Urban renewal (using the lightning bolts as a kind of wrecking ball end) is a possible downtime activity for this spell caster. How do the local activists who want to save their neighborhood stop this destruction? Break the spell caster's concentration!
Rule of Cool?
The Arcana contest sounds like a fun ruling! Both spell casters try to concentrate on the storm, but There Can Be Only One who calls down the lightning!
