I think the answer is maybe yes and maybe no.
They do sometimes drown in the torrent but often make a remarkable recovery and haul themselves out of the water and when quiet enough, he/she may come out of the overflow (which is conected to the waste pipe) and get trapped in the bath again. If one keeps reappearing, its a pretty safe bet thats whats going on.
If you can pluck up the courage, its best to release them outside, in my opinion. A humane way of doing this is to place a glass over the top of it, then slide a piece of card under nieth then take him on a trip to the yard, a piece of cake once you've done it a couple of times!
Another method is, scrunch up some toilet tisue and place it over the spider and gently squeeze your fingers together to hold him until you get outside.
NOTE, This applies to non venimus spiders in the UK.