I'm thinking about painting my basement's walls with some concrete sealer (before finishing the basement) and floor with epoxy paint. Many people are doing that, but I'm concerned about moisture in the concrete. The outside of the walls are covered with some black stuff which is a moisture barrier. If I paint the inside of the walls, I will lock moisture inside the concrete for ever - will it affect concrete in any way? What about mold in the concrete? I heard something about concrete breathing - does it apply here?
Yes, you will trap moisture with the epoxy. Will that hurt? I'm not sure.
That said, it sounds like your foundation walls are already waterproofed on the outside...which is great, as that is where you want to do the waterproofing. As such, I wouldn't bother doing it internally.
And if you are interested in how to finish the basement, I put a detailed answer here: Should I use steel or wood studs for basement exterior walls?
Would moisture be trapped? To some extent. Is it bad? No, once concrete has been mixed and started to cure, keeping it moist helps it get stronger and this effect can go on for a long time. In no case will it ever make the concrete weaker. See the graph at the end of this URL:
There are some who say the concrete in the Hoover Dam is still curing.
There is no need for concrete to breath from the point of view of the strength of the material. There will be moisture retained from the original mix and it is possible some are confused about the chemistry of concrete and think it needs to dry, but the truth is the complete opposite of that.