I'm specifically interested in paid rowers, though I'm not sure how different their experience would have been from slaves (if there is any distinction that you know of, I'd love to know!)

The difference would be that, with few exceptions (I can think of exactly one off the top of my head), there were no slaves rowing warships in ancient fleets at all. In fact, it's a defining feature of the nature of naval power in antiquity, that it has enormous political repercussions for most states to maintain and man a significant permanent fleet. Rowers were free, and in the few cases where slaves were used to row (due to shortages of manpower) they were either freed beforehand or promised their freedom afterward.

Generally speaking rowers on board warships didn't actually do much of anything for most of the time. The multi-banked galleys of antiquity were relatively poor sailors, being designed for battle more than seaworthiness, and while they could travel much faster under oars they generally put up their masts and traveled by sail outside of battle, since no crew could last very long under oars, it was too exhausting. As a result the majority of a rower's time during a journey would be spent kinda just sitting around below decks at his bench. Warships were generally beached at night and during mealtimes--these ships couldn't be trusted not to capsize at night, and as weapons first and ships second they didn't have space for food. The crew would therefore have to physically beach the ship and either forage for food or buy it if they were at a port--at Aegospotami Lysander was successful because he managed to locate the Athenian fleet while they were off buying food. In battle it's hard to reconstruct a rower's experience. A rower could not see much of the battle, or much of anything really. He was crammed into a cramped space, which could get quite unpleasant if he was on the bottom, and his contribution to the battle was to row as strenuously as he could--in time with his comrades of course. The rowers of an Athenian ship were subdivided according to their bank, with the topmost bank receiving a pay bonus (since their work required the most physical effort). The deck-rowers were not the most unfortunate, however--that distinction must go to the bottom bank. The lowest bank of rowers had the unpleasant experience of being dangerously close to the water, and of being...on the bottom--Aristophanes makes a joke about how when you're on the bottom bank you have to deal with everybody above you farting right into your face (and it really is kind of an ass-to-face thing, the rowers were pretty literally crammed in there). Besides this we know fairly little about a rower's experience in battle, although there can't have been all that much more to it, he couldn't see anything. Rowers appear to generally have sat by during boarding actions, since a small force of marines was usually enough to take a ship if its own marines and crew had been dealt with. Rowers could be armed sometimes, however. At Pylos Demosthenes armed around a thousand rowers with makeshift wickerwork shields, and successfully used them along with his handful of marines to defend the promontory from the Spartan army. Likewise, rowers probably participated in the raids the Athenians made on the coast of the Peloponnese, although we should probably expect that their contribution was more in the way of burning shit than actually fighting