Since inspiration is binary (a PC either has it, or not--no stacking), and as players can give it away to other players, I would lean toward as often as possible.
And if I were playing a bard or warlord (fighter), I would work towards gaining it as often as possible to give away to the other PCs.
Your players -forgot- to use inspiration? How could they turn down a free "+5" to a roll?I'm curious to how often those DMs that are playing hand them out and what criteria they use.
Yes, it's weird for players to be giving each other bonuses. But the same thing happened with Aid Another actions. Look at it this way: players who don't already have inspiration are probably not the types to be earning inspiration. If the inspirational PC hands his inspiration out to non-inspirational players, there's not going to be an unbalancing amount going around, since only part of the group is the type to be earning inspiration. (Insert better wording of this concept after this post, thanks.)To me, it's kind of odd that WotC gave the players the options to distribute a metagaming tool like this, but meh. I can always control it.

(Dungeons & Dragons)
Rulebook featuring "high magic" options, including a host of new spells.