Yes, the joists are probably pocketed into the brickwork, and there are probably gaps. You would to take up at least some of the floorboards for access. Clean out loose material by scraping and using a builder's canister vacuum (not a domestic vac). You can pack them with sand and cement mortar but you will probably find expanding foam easier. Remove all the dust and spray the hole with water to make it stick. Use the plastic extension tube to get it deep into the hole. I use the pink fire foam which blocks smoke and flame, it chars but does not burn. Most likely you will never have a fire, but, why not?
In some cases there is a separate sleeper wall just inside, but I think I have only seen that on exterior walls where it protects from damp.
Gaps in the brickwork are worse in unplastered areas ( plaster hides them ) between floors and sometimes inside cupboards. Builders traditionally used the best bricklayers for the exposed external walls at the front, and hidden brickwork is often very shoddy.
Afterwards, you can pack between the joists with dense mineral wool which muffles noise (loft insulation is much lighter, so not as good).
The relevance of the chimneybreast is that the wall, between your fireplace, and next door, may be very thin, just a flue, and this is not as good for blocking sound. If it was just plastered, plaster is less dense and not as effective as dense thick brickwork.