It's not wise to give a publicist a cut of your royalties and I've never heard of them being paid that way. A fee is the more usual way to pay them: this can be a flat fee, a retainer plus specified extras and expenses, or a fee per task--although an overall fee plus reasonable expenses is most usual.

A publicist is hired to generate publicity for your books. So, they might arrange radio, press and/or TV interviews for you, a blog tour, book signings, or talks to relevant organisations. They'll talk with you first and work out with you the things you're happy doing and the things you'd rather not, and between you you'll work out a plan.

None of this will be effective if the people who see the publicity generated can't then easily buy your book: so if you engage a publicist who arranges an interview for you with a local radio station, for example, you have to ensure that your book is available in local bookshops and related retailers when that interview will be heard. That part of the work isn't usually down to the publicist, but is essential if the publicity generated is going to be effective in selling your books.