This would not really work because, for the most part, fire protection is a public good. It is not really possible to provide it to one household without providing it to others.
It might be possible if all the houses were single-family houses and if they were so far apart that a fire from one could not possibly spread to another.
But in a city, it would be impossible. You can't sell fire protection to Apartment 101 and deny it to 102. If you save 101 from burning, you're going to save 102.
So, a fire department like the one you describe would have a hard time selling its services due to the free rider problem.
There have been privately run fire departments in the past; there are quite a few now. It seems easy to provide fire emergency and protection services on a renewable contractual basis for a small town that is paid for by local property taxes. There are a great deal of privately run fire rescue services which provide advanced and cardiac first aid. It is generally a concept that towns and rural areas look to in a downward economy, and where public fire departments just can't get to or find it extremely difficult. Many towns and small cities just don't have the interest from people to form a volunteer department, so the next best thing is to hire a private company.
I tend to agree that it would be quite hard to regulate. However, one scenario that might work is to supplement an existing state-run department. Here on Long Island, we are separate from NY City. Our FD's are all volunteer. The men and women who join get certain benefits, like a large tax break on their property taxes and if they work volunteer long enough, they get a pension. But often times, they are short on manpower. Now, think of a private FD that could piggyback or support the local volunteer FD at times. The private FD could bill the local town or state gov't to pay. I could see that scenario possibly working.