All of them, except maybe 4e
In spite of AD&D 1e infamously having a table for prostitutes, D&D in general avoids explicitly discussing the subject of fornication. So we'll have to derive this from fundamentals.
What can the shape do?
All editions share that the body of the druid becomes that of the animal, in 1e "in all respects save the mind", in 2e "all of the creature's characteristics", in 5e "your game statistics are replaced". There is nothing in the text that would exclude any of the natural functions of the form, in any of these editions. Restrictions are typically around special benefits from class or race (like spellcasting or darkvision) that do not automatically transfer. 4e seems to be the exception here, you change shape but retain your old characteristics.
Procreation is a fundamental, normal biological function of any animal. With text stating you take on the physical characteristics of the creature, and no text excluding it explicitly, there is no reason whatsoever to believe that the shape-changed form should be unable to do it.
The only challenge here is that the subject itself is something people might not be comfortable with. If so, you probably should avoid the subject. If however this is something you discuss for fun in a pub, as seems to be the case here, that should not be a problem.
What's left behind when you revert
In 5e it's not clear from the rules if removed parts of the wild shape body persist after the druid leaves wild shape. The same is true for older editions.
So any insemination might vanish when you revert back, depending on how your DM rules. This can pose a practical challenge for creating offspring.
If the DM rules that spilt blood, spittle, lost feathers, cut hair and so on are not part of the body anymore, then you can procreate in all of the editions (except 4e).
If the DM rules that all these are still part of the form you took and will revert or dissolve when you revert to your normal form, this will undo any effort, unless you manage to stay in wild shape long enough.
Staying in shape
In older editions there was no cap on the time you could stay in wild shape. In 5e, you could as a high-level druid: from level 12 on you can stay in shape for 6 hours, enough for sleep in a long rest, and with two uses per long or short rest, you can keep it up forever if you take at least one short rest per day.
While in 3.5 you can use Wild Shape multiple times per day at higher levels, unlike in 5e, the text does not state you can stay in form. But in 3.5 detached body parts retain their shape, so it may not be an issue to begin with.
That would mean you could stay in shape long enough for the child to be born, even if the original cell you gave then vanishes when you finally revert: at that point it should not matter anymore, billions of copies have been made.
PS.
If you change into a bear, and sire a bear cup, because all of your bodily attributes are that of a bear, the cub would be a normal bear. It would not gain any of your mental abilities.
Rules Text
In 5e:
Your game statistics are replaced by the statistics of the beast, but you retain your alignment, personality, and Intelligence, Wisdom, and Charisma scores.
In 4e:
Wild shape allows a character to change between humanoid form and beast form. While in beast form, a character can use powers with the beast form keyword, but cannot use weapon or implement attacks lacking the beast form keyword.
You can choose a specific form when you use wild shape to change into beast form. The beast form is your size, resembles a natural beast or a fey beast, and normally doesn't change your game statistics or movement modes.
In 3.5:
At 5th level, a druid gains the ability to turn herself into any Small or Medium animal and back again once per day. Her options for new forms include all creatures with the animal type. This ability functions like the alternate form special ability, except as noted here. The effect lasts for 1 hour per druid level, or until she changes back. Changing form (to animal or back) is a standard action and doesn’t provoke an attack of opportunity. Each time you use wild shape, you regain lost hit points as if you had rested for a night. (…) A druid can use this ability more times per day at 6th, 7th, 10th, 14th, and 18th level
In 2e:
Each animal form (reptile, bird, or mammal) can be used only once per day. The size can vary from that of a bullfrog or small bird to as large as a black bear. Upon assuming a new form the druid heals 10 to 60 percent (ld6x10) of all damage he has suffered (round down).The druid can only assume of a normal (real world) animal in its normal proportions, but by doing so he takes on all of that creature's characteristics-its movement rate and abilities, its Armor Class, number of attacks, and damage per attack.
In 1e:
Ability to change form up to three times per day, actually becoming, in all respects save the mind, a reptile, bird or mammal.
- Each type of creature form can be assumed but once per day.
- The size of creature form assumed con vary from as small as a bullfrog, bluejay or bat to as large as a large snake, an eagle, or o black bear (about double the weight of the druid).
- Each assumption of a new form removes from 10% to 60% (d6, multiply by 10) of the hit points of damage, if any, the druid has sustained prior to changing form.
In OD&D:
Druids were introduced to the game as a monster in the Greyhawk supplement, and made a playable class in the Edritch Wizardry supplement. It states (p. 2)
Upon reaching the 5th Circle druids then gain the power to
shape change (as previously mentioned in GREYHAWK with regard to the Druid-type monster)
and the referenced Greyhawk supplement's rules for it are:
Druids may change shape three times per day, once each to any reptile, bird and animal respectively, from size as small as a raven to as large as a small bear
There is no further limitation on duration.
