It's something of a nebulous title really.
The functions of a producer vary greatly - some simply oversee everything, book the studio, get everyone together in the right place at the right time.
Others have a large say in musical arrangements, choice of tracks, overseeing mixing and studio production generally, so it does depend on the individual and the band / artist they are dealing with.
For example - Mick and Keef have produced The Stones albums for years, and probably do all of the above. Someone like Bob Clearmountain is known for mixing sound in a specific way, with an identifiable sound at the end of it.
So there are no hard and fast rules really - hope this is not too vague - but then again, that's how vague the job spec. is.