I've been a paparrazzi - and I didn't stalk a soul.
The term is used to cover everything from the classic stalker photographer that hides out and uses long lenses to celebrity news photogs (which is what I did). The news photographers are part of the game - make no mistake about that. They shoot movie openings, events et cetera and the celebs know full well they will be there. There is even a pecking order of photogs decided by the celebs' publicists as to who gets to stand where in the photo pit. A new photographer has to be approved by the publicists. You get a wrist band and everything.
TMZ hires a lot of young people for the job. In large cities, photo agencies will hire young photogs. Many of them do both celebrity news and stalkerazzi work. Some of the agencies that had photogs at the jobs I worked: Getty, Starz, Wired - these are hard news agencies and you will get more respect if you can get hired on here. They don't pay people to stalk. There are a lot of pap agencies as well, and they'll hire you too, but the other photogs may give you some grief. These are the stalker agencies. Google photo agencies in your area. Works best in LA, NY and London.
Understand you don't get paid a penny until the photo sells. You are treated as a contractor and are responsible for your own taxes-social security as well as income taxes. See an accountant. Also, understand that if you do this, especially if you work for a stalker agency and get "good" shots, you will be painted with the brush forever and may have trouble being taken seriously as a legitimate photographer in the future.
Many celebs have someone on their staffs inform the agencies where their star will be on any given day. The celeb arrives at the store/airport/mall/where ever, the photogs shoot and then everybody is done with it for the day. The smart ones sell photos directly to the tabloids so as to avoid a lot of the bs and the celeb gets the $ (Julia Roberts did this with her kids).
P.S. Like drugs, these kind of photos will always exist as long as there is a market.