I can't fully guarantee this info since all that I know is heard from friends who didn't actually went hunting. One sure thing that I know is in Romania bow hunting is illegal.
However I heard that the closest area from my place where one can bow hunt is Bulgaria. They say you have to have a IBEP certificate. Unfortunately although I live 30 miles from the border with them I can't understand a single word in Bulgarian since they speak a slavic language and - like the Russians - use a different alphabet from ours. The most notable game out there I guess should be wild sheep, jackal, boar, fox, rabbit, deer and pheasant.
In Hungary also I've heard that one can bow hunt after taking a local exam. It should be recommended since the Hungarians take anti-poaching measures quite serious so you could actually encounter some wildlife out there.
Game: boar (they have a strong tradition in boar hunting with hounds), fox, rabbit, pheasant and deer.
In Slovakia also it should be legal with IBEP certificate.
In Ukraine and Russia my guess is it might be legal and they should have a great variety of game species including brown bear and elk.
In the rest of Eastern Europe to my knowledge bow hunting is permitted only on private property inside of a fence.
In general there are not so many invasive species in the wild out here and they don't present a tremendous interest for the bow hunter - like some tortoises and some species of very small fish.
One should also know the fact that the game of Eastern Europe is also heavily poached with various methods raging from slipknot traps to suppressed fire arms.