What are the qualities of a successful beta testing session from the player's perspective?
Designer and sometimes attendee at design group BOGA DAP here. I'll share my personal experiences and preferences.

As a tester, to feel good about the event:


In general, I prefer the game to be taught as if you were teaching an already-published game. If you are getting a blind playtest, or testing the ability of players to learn from a rulebook, then I expect the game to be 99% done and the instructions to be as clear as you can make them.
If you have specific questions about the game you want answered, in terms of balance or how a mechanism works or how it feels or whatever, let me know. In a playtest game, I am not averse to you changing the rules mid-game if you want to try something out or the rules aren't working.
I have no expectations at all in terms of graphic polish or illustration. An unpolished prototype is a sign that you are not emotionally invested in the game as it is and are open to feedback and changes. Even going so far as a deliberately sketched-in look-and-feel can be helpful.
I expect you to respect my time. I play games for fun all the time, so if the game is fun or at least interesting, I don't need a food bribe. I do want you to be willing to playtest my games as well, and I want you to make appropriate changes or pull the plug if it ends up not working or not being fun.
1-2 hours, with time on the end to ask for feedback, makes a good playtesting session for most games. An early playtest (where we expect to pull the plug quickly and go back and make changes) might end up being shorter, or a test for a very short game. I don't expect to play a medium-or-longer game more than once in a sitting. Unless the game was a smash hit, I expect you to take it back for at least a few changes before we play again.


As a designer, to get the most out of a playtest:


Try to observe the game, rather than playing it yourself, unless you are needed to fill up the player count. This lets you pay more attention to both the game and the players and how they're acting and feeling.
Tell the players what you changed since the last iteration of the prototype. If it's not obvious whether it worked or not, ask the players.
After the game, ask each player what they liked and didn't like. This is tricky. On one hand, never defend your game. If a player gives a criticism that you feel is invalid, ask yourself what you might be able to change to mitigate the perception of the problem - whether in the rules or the explanation. On the other hand, you will need to pass by many suggestions, since they will not lead your game in the direction you want it to go in. Not every game is meant for every player. Criticism might just be telling you who the best audience for your game is.