The impact varies with the composition of the two parties
This answer is informed by considerable at table experience across a number of editions. But first, Icyfire makes a significant point that I need to repeat. Side Initiative (DMG p. 270) disadvantages PC's with high initiative bonuses. (We saw this with Monks and Rangers in OD&D and 1e, see below).
One thing to consider is that some of your characters might have some
boost to their initiative rolls through class features (barbarian) or
feats (alert), and using this initiative system makes those bonuses
worthless. This is especially pertinent for the DMG variant, which
explicitly states that no bonuses are added to any side.
Some people have found this to make a given encounter one sided.
Higher level
(@Evoker) I'll give it a huge warning. At high levels a group going
last can mean they're all dead before they get a turn. Sure, the
players love wiping out the enemy without a scratch, but aren't as
happy if it goes the other way. The action economy swings wildly if
it's an entire side acting before another.
Lower Level
(@Doval) I've had issues with this at low levels as well, because the
normal initiative rules say you roll once per group of identical
enemies. The Starter Set has encounters that are just 3-4 of the same
low-CR creatures, and having them gang up on a single level 1 or 2
character often left them almost dead with no chance of other PCs
intervening. I can see this problem getting much worse if mixed
monster groups with complementary abilities get to act together
My own Experience, 1
In OD&D we found that side initiative (and surprise when it occurred, which was usually rolled for) coupled together to make for lethal first rounds. Who won initiative (and surprise, if any) made a huge difference in how the combat went.
What my first DM did was create an event flow of simultaneous
outcomes which worked like this: unless there was surprise (and with
a monk in our party, he always rolled separately) the DM would do the
monsters/NPC's, we'd roll the character actions, and all
results/impacts would count. (that means my dwarf and the orc across
from him could both kill/down each other during the same round).
The advantage of this system is that you'd not miss a chance to swing or to cast just because the enemy hit you, unless surprise was involved. It worked pretty well, and still made setting up surprise a big force multiplier -- we put some effort in trying to surprise enemies.
At other tables, we had more than one party die in the first fight in a few OD&D games due to the foe having initiative and a few warm dice ... and the snowball would grow fast once the first few fell ...
Empire of the Petal Throne: as DM, I tried various modes of
initiative, and we settled on the "simultaneous outcomes" idea my
first DM had. All three tables I ran that on liked it, though as we
got up in levels a lot of the advantage were still ranged attacks and
surprise in nature, and using both spells and terrain to control the
fight.
Because we rolled for both, some classes having a benefit on surprise got less advantage with a side initiative. (Monk, Blackmoor, p. 1)
At 3rd level monks are surprised only on a roll of 1 in 6, at the 5th level only 1 in 8, and at 7th level and above only 1 in 10.
My own Experience, 2; when things got all fiddly
AD&D 1e: we tried a lot of different systems in this edition, and
found once again that side initiative, particularly when coupled with
surprise, made a wipe (either way) more likely. We spent a lot of
time with a segment-based system that Len Lakofka published in
Dragon magazine. (Using a 1d10 and various adjustments to see when things happened during the round, spell casting times, quite fiddly).
We found that if missile equipped foes had the initiative, most
casters on the opposing side were down before they got a chance to
cast. Low HP casters contributed to this.
More surprise / penalizing a class with a nice feature using side initiative ...
(1e PHB)(p. 25) Rangers surprise opponents 50% of the time (d6, score 1 through 3) and are themselves surprised only 16 1/3% of the time (d6, score 1). {snip} p. 30 Monks at 1st level of experience, a monk is as likely to be surprised as any other
character, i.e. 33 1/3%. {1 or 2 on d6} This chance goes down to 32% at 2nd level, and it thereafter goes down 2% per level, so there is only a 30% chance of surprising a 3rd level monk, 28% chance at 4th level, 26% chance at 5th
level, etc.
Note: since we rolled for surprise and initiative in a lot of combats, this is similar to the current initiative roll since surprise is no longer rolled for, and it makes that one initiative roll an even bigger deal.
AD&D 2e: Mostly played this with side initiative, and going first was
a distinct advantage. As I got very little high level play, cannot
report on how that changed a later levels. Did not play this nearly
as much as 1e.
AD&D 3.x: not enough play to comment, combat too fiddly, also
difficulty getting games to stay together due to IRL. 4e: didn't
bother to play. No comment.
In the Current Edition (5e)
As our group has gotten used to this system, we are taking more advantage of the action economy and are finding that making the most of the mechanics makes combat encounters a lot more fun, even though a lot of the appeal for us in this edition is the KISS principle. Initiative in this edition is a little fiddly, but very workable. We experienced a learning curve. Last year, as I was messing around in roll20 with DM tools, I ran a number of mock combats (with my son playing various sides, using varying numbers of creatures on both sides with Party versus NPC/Monsters as mixed below). Add into this high and low mixes of missile versus melee combatants. DMG variant (p. 270) side initiative was something we looked at. Side initiative didn't appeal to us as much as the system in the book.
Small-small. Small-large. Large-small. Large-large.
We used up through CR 7-8 encounters. The small-large, and large-small encounters showed this most prominently, but even large-large showed it somewhat. (Heh, the young green dragon getting initiative and a breath weapon wasn't pretty. :p) )
I found the same kind of "single roll" advantage with side initiative that I'd seen before, particularly when you added in bonus actions and things like the goblin disengage feature and hobgoblin martial advantage, Roper grapple hits ... web spells ... obviously AoE abilities. The ability to overwhelm (fight snowballs quickly in a round or two) the side that goes second, particularly at levels 1-3 where players start off, is not trivial. This edition' action economy was built and balanced (or such was attempted) with variable initiative in mind, not side initiative.
While I do not recommend side initiative in this edition, I understand that once the swingy nature of low level combat is mitigated by higher HP and chances to shape the battlefield, that particular concern may not predominate.
The other benefit we found to using initiative as written is that it forced you to make a choice if all of a sudden the bard went down right before you were thinking of making a charge move: maybe you instead defend him or something else. Our group finds that an appealing feature, but perhaps not all tables like that kind of pressure/decision making.
Recommendation 1: wait for level 5 or later PC's to do this, if you do it
Don't do this at low levels. Wait until at least tier 2 (levels 5 and up) to try side initiative at your table (particularly with new players) once they have gotten used to the basic system. Combat outcomes are swingy at low levels due to there being crits, d20 rolls, low HP at low levels, and most monsters have +1, +2, and even +3 on damage die.
For low levels (tier 1), I'd recommend to play it as written. Break up sizeable groups of monsters into clumps of 3, 4, or 5 to have common initiatives as DM -- this simplifies your workload. (Leaders always get their own initiative).
Recommendation 2: Listen To Your Players!
If your table wants try the optional rule, then use it for a few sessions and then have a sit down and ask about their likes and dislikes. Make sure you've done numerous encounters/combats so that your group is not the victim of small sample sizes. I'd say ten, but more would be better. Find out what the consensus is, and go with that.
Whatever works best at your table ... do that!
I just found this answer by @Matt Vincent that is similar to side initiative, that he's used for years, and you may find it a happy medium.