Transposition Tables Are transposition tables (in engines) used for one or multiple games?

My current knowledge of transposition tables is that when an engine searches the game tree, there will sometimes be positions that have already been calculated earlier. A transposition table allows an engine to take what it already had calculated earlier and use it for even deeper search.
What I don't understand is whether transposition tables store positions permanently (such that an engine can use what it previously calculated in previous games for new games) or if it's strictly game-specific as in that a new game corresponds with a completely empty and new transposition table.
When you shut off the engine and turn it back on, its transposition table has been cleared. In order to continue using the saved positions, a database of some sort would be needed. This would keep getting bigger and bigger the more you used the engine though, and could become very inefficient for the engine to actually use to the sheer size.
If by a new game you mean going back to the starting position and playing out new moves (all while keeping the engine on), then it probably depends on the engine. Some engines might clear out their transposition table in order to free up space, since the old saved positions happen after a different series of moves. Engines can't keep using more and more RAM, so they have to manage their TTs effectively and get rid of positions that likely won't be useful soon/ever while the engine remains running.