I had a very bad experience with a colleague and just do not like/trust him. (Eavesdropping, lying, ...) I am working on a project with my mentor, and know they shared information with the colleague I do not trust. How can I politely point that out to them?
If you can trust your mentor, just tell them. Say it as it is. But not in an email. This requires personal communication.
But, if you cannot trust your mentor then you have a bigger problem than you state and need to find better guidance.
As a meta comment, sharing your research is on average the best way to progress in my opinion. Sure, there are some bad apples out there, but overall I feel it’s wise to err on the side of collaboration and communication.
I suggest that my students prepare human readable manuscripts that they can share with others. No one stole their work yet (though it did happen to me once). Again, on average this is better I think.
To your question: I think directly and politely asking is the best way. “Hi, I feel like the work is not quite ready to be discussed as it’s too preliminary, would it be ok if we develop it more before we present it? I feel it would have more impact this way”
Don’t mention distrust or plagiarism. If your professor is not completely dense they’d understand that this is part of it.
This situation will require you to go see your mentor in person, be friendly and straight forward. Just let him know how you feel.
Much depends on HOW you say something like this.
Of course this must be expressed in person, but if you tell your mentor directly, as in "sir, can I ask you to do one thing, ... etc." then there is a likelihood that it won't be taken too seriously. Why? Because, indirectly, you are telling your mentor that you understand things better than him/her -- that you understand this other person is untrustworthy, something the mentor didn't realize by himself. Human nature is that nobody likes to be corrected.
So, in my opinion, a better approach would be in the form of asking the mentor for advice.
Instead of just telling him what to do -- "please don't talk about this in front of others", ask him advice on how to handle the situation. "Sir, can I ask you one thing?" (sure...) "Maybe you will feel I am a bit silly..." (no, don't worry, what is it?) "Thanks... well, the fact is that I noticed that after our collective discussions, person X is doing A and B and C and this is a problem for me because D and E, etc." (um, I see...) "So how should I handle a situation like this?" -- at this point, if you made your case convincingly, your mentor may "spontaneously" come to the right conclusion: (well, you know what, maybe next time you should talk to me when they are not around).
(By the way, do not badmouth this other person unnecessarily. Be fair and balanced while making your case)
Bottom line: if you can make the mentor feel that he made the decision, and at the same time let him save face (for not noticing the problem themselves) then there are good chances that things will go the way you want, and not just temporarily but permanently, especially if the way you want is also a good way which in the end benefits everyone.