Often recording musicians might be shown an outline of a song: the melody, chord structure, perhaps a sketch of what ideas they would like to include on the record (played or sung to them) and the musician is expected to work from this. In many (most?) cases today, sheet music is not used in the session at all; songwriters aren't writing out exact parts for what each instrument is doing (especially for guitar parts).
That's why it's incredibly important for a session musician to be able to audiate and know their way around an instrument. Anything I sing you, you should be able to play back to me. Depends on the artist of course, in some cases an artist might have a very clear idea of exactly what they want on the record, but in other cases it might be more vague and part of the session musicians job is to fill in the gaps around the ideas they're given.
The other thing is that the semi-compositional role of producers is growing an growing. In modern recording sessions, some producers just direct a musician to play certain things in a certain way, and do a few takes until they get something that they can work with. When the session musician finally hears the end result they might be surprised to hear a part they never really played; it's a collage of the musical ideas they came up with in that environment that's been artfully stitched together by a producer.
The point is, the main skills required to be a session musician are a very good ear, ability to improvise and interpret music and of course technical ability on the instrument. That being said, reading music will still open doors for you and if you are looking at a career as a session musician, you should probably learn to read (even if you are not a sightreader).