When we go swimming and spend extended time in water, do the little bugs who live on our skin suffocate and die?

As another user pointed out, most bacteria and other microscopic forms of life don't really "breath" in the way we think of it. Often they just absorb oxygen from their surrounding environments.

They can be washed away, killed by chlorine (pools), or killed by salt imbalance (ocean). However it's really hard (read near impossible) to kill them all, and even if you did they exist in our environments naturally and from other humans as well as on things we've touched recently. So they repopulate quickly.

There are some arguments that Sodium Laurel Sulfate kills "good" bacteria on our skin.