Of course there are hunting seasons (there are two of them: one in spring and the other one in autumn), even for duck hunting. And for large animals (like boars, bears, etc.) there is a strict quota. As far as I know it works like this: say, a company of 5 hunters wants to shoot a bear. So, they must buy a ticket for that (and it's rather expensive). And there will be written in this ticket that they can shoot only one bear. If they would shot two it would considered as poaching.
Ukraine and Belarus, as European part of Russia are rather dence populated, so I think today you can meet a bear there only in national parks or so. But at sparsely populated Russian North, Siberia and Far East regions freely roaming bears are still a commonplace.Are there bears in Ukraine and Belarus too?