When you're riding the bike, can you still coast without pedaling?
When pushing the bike, do the pedals spin while in any gear, or only some gears?
Is the bike old or new?
I suspect the friction to spin the chain/cranks has become less than the friction in the freewheeling mechanism. Thus the bike preferentially spins the pedals. The freewheeling mechanism (whether a freehub or freewheel) can potentially be freed up by dripping some oil into it.