In general, with my current language partner we only correct when:
1) It was so bad as to be almost unrecognizable. i.e. "Did you mean xxxx?"
2) If it is a common mispronunciation that is being done often. The best way to correct this is to respond using the correct pronunciation so they can hear it used in context correctly. i.e. you don't stop and say "you said that wrong". You continue the conversation but work the word into your next response. Example: "She came wit me". Response: "My wife came WITH me as well". Perhaps with a slight overemphasis on the word in question, but don't over do it.
3) When directly asked. i.e. "How do I pronounce xxxxx correctly? I have trouble with that."
In general, you don't want to break the flow of the conversation with lots of corrections. Occasionally it's necessary when you can't tell what the other person is saying, but usually we try to correct by example and stay in the conversation.
It's very important to remain respectful and not get offended when you are corrected. If both parties are able to laugh at their own mistakes and treat it as a learning opportunity then it helps a lot.
If you are worried at all about how to correct each other you should discuss this before your first exchange. Setting expectations and some basic ground rules before starting can avoid a lot of misunderstandings and/or hurt feelings. We are not telepathic and mixing in foreign languages can create a recipe for miscommunication.
In my experience, native English speakers usually do not correct your pronunciation unless they don't understand you at all (and even then, many would just smile hoping you didn't say anything important). Should they? Well, it's a difficult question. You certainly can't ask your friends to correct you if they haven't explicitly agreed to it, it would be just irritating. It's supposed to be easier with language partners (in theory, both of you are well-meaning and want to improve your languages), but I've noticed that many people genuinely feel hurt and sad when being told about their mistakes.
Vince mentions responding using the correct pronunciation but personally, I prefer directness and honesty. I suppose I'm just not smart enough for this subtle approach because most of the times someone does what Vince described, I end up wondering what exactly was wrong with my sentence. Was it grammar? Choice of words? Intonation? Pronunciation? What exactly was wrong? Funnily enough, when asked directly, some people would just politely say something like "Oh don't worry about it, please go on, you're doing great" (what's the point then?). Needless to say, I never feel confident enough about correcting them because if they think I would be hurt because of such minor things, how can I be sure that saying "Oh, it's actually pronounced "ят", not "яд", we normally substitute word-final /d/ for /t/ in Russian" won't discourage them from learning the language? There are people who don't want to be corrected at all, after all. To each their own, I guess, so it definitely should be discussed beforehand.
I'm English and I do help with pronunciation, I can do this because I have learnt other languages and how to explain certain differences between languages.
When I speak Spanish/Deutsch in my experience I DO get corrections, sometimes small ones, because I worked at pronunciation so it's not overwhelming. Of course I have an accent, but if I mess up, they regard it as comment worthy.
In my partners, understandability comes first, but I do explain how to speak English more clearly.
That said, I try and listen a few minutes, then decide what my exchange partner can improve with the biggest pay off. Often professional teachers are less picky, probably because they're used to worse students.