The amount will depend on the size of the publisher of course but they have a vested interest in high book sales. For there to be high book sales, the general public needs to be aware of a book and feel the need to read it.
The first thing I suggest is getting in writing whether through Email or directly in your contract the detailed steps the publisher takes for publicity. This is called a marketing plan. Get it in writing as this is insurance for the future if you find out they aren’t doing what they said they would do.
If the publisher asks for a list of your friends and family to notify about your book,then expect little else publicity-wise from them. This is the first sign you will be doing a lot of your own legwork and you should consider an outside publicist. Honestly, you have probably already told everyone you are publishing a book. Why do you need the publisher to duplicate your efforts?
If they tell you they are contacting your media, make them be specific in writing exactly what their efforts will be and how many hours they will spend on your book.
A particular Publish-on-Demand company declares they are a “traditional” publisher but the only media contacts they perform is sending a flyer to your local media notifying them of the book. There is no follow up. There are no targeted media groups contacted nationwide. Most authors find themselves told months after publication they should consider hiring an outside publicist if they want to see more sales. By that time, it can be too late for a truly successful publicity campaign.
Ask for the marketing plan the publisher developed for your book. A commercial “traditional” publisher will at least have a general outline of marketing efforts created before they officially take on your book.
Large traditional publishing houses are promoting your book months before a single official book is printed. They send out advance review copies or galleys to the top reviewers like The New York Times and other venues targeted to your book’s topic. If your publisher isn’t sending out free galleys for publicity months in advance of your release date at their own cost, they aren’t a truly traditional commercial publisher.