It’s a difficult question to answer.
Is it possible for someone, just wearing a mask (made of latex or whatever), to successfully disguise their own appearance without it being obvious they are wearing a mask? I don’t think so, and I’ve never seen it done or heard of it being done successfully. People can disguise their own features to conceal their own identity, but this is not done by wearing a mask.
Is it possible for someone to successfully disguise their own appearance using the services of a special make up FX team? Yes. The process involves taking a life cast of the person’s face, then building a ‘new face’ by creating a rubber, foam or latex mask or set of mask pieces which fit on to the life cast. The mask (or these separate pieces) are then glued on to the person’s face, and everything is blended together using several layers of make-up followed by a very careful process of surface painting, often inlcuding an ‘air brush’ type of applicator to produce subtle tones and a very high level of surface detail and variation (because real skin is never completely pure or even in tone).
That this is possible has been demonstrated many times. There was a British TV show a few years ago called “Celebrity Swaps”, in which some very well-known people underwent this process. The celebs were transformed as above (a process that could take weeks to complete), and then met and socialised with people who knew them in real life, to see if their friends would spot who they were. In most cases, they got away with it. In one case, a famous model and TV presenter (Melinda Messenger) was not even recognised by her own mother. In another case, a man called James Hewitt, easily one of the most recognisable people in Britain through his association with the late Princess Di, passed completely un-noticed even when among people who had known him most of his life.
Note that this is not quite as specified in the OP, in that the transformation does not usually involve a single-piece ‘mask’ as such, but several prosthetic pieces which are blended together. Also, there is much more to the transformation than just the mask or mask elements - there are several other make-up processes involved. It should also be mentioned that in order to disguise themselves, the celebs also had professionally designed and applied wigs, clothing and costume advice, acting lessons and lessons in how to moderate their voice and speech patterns.
Note also that asking if person A can hide their own identity is quite distinct from asking whether person A can successfully impersonate person B, using a mask or special FX make-up or anything else. I have not heard of anyone doing this successfully, except in cases where the people who were fooled didn’t know what person B looks like. In other words, it’s easy to forge papers and official documents and pretend I am the Ambassador From Timbuctoo, provided the people I am trying to fool don’t know what the real Ambassador looks like.
In the original ‘Mission Impossible’ TV show, the characters often impersonated other people by using single piece latex masks that they could put on or take off at will, and this worked even among people who knew the impersonated person very well. This was, of course, fiction, and it was plainly implausible. There was often a laughable disparity between the size and shape of the heads of the two people, so that even a very good mask would not have been usable in this way.
So, the short and sweet answer to the OP is: no, it can’t be done the way they do it in the movies with a simple ‘rip off’ mask. Modern special FX make-up can enable you to disguise who you are, even if you are sitting among your best friends, but this isn’t the same as successfully impersonating another person.