The down votes are about not having done due diligence.
Private investigators don’t break stalking laws because they don’t need to. They can do their jobs comfortably, from a distance, without any illegal acts at all.
For the question in the post itself, they don’t even need to have anything to do with the subject of their job. They’ll be looking into backgrounds, digging through records as their main method. Most of what any PI does is digging through records.
Even when they observe and record current activity of a person, there’s no need to violate stalking laws the get the job done, because those laws are written with pretty specific language. With the caveat that the exact wording is going to vary by jurisdiction, there’s no sustained harassment, or any direct contact at all during a normal PI job. Even photography or other records are made from a distance, and in places where the expectation of privacy doesn’t apply. More important, they don’t sustain that level of surveillance past the point where the case ends.
Now, that’s not saying that individual PIs don’t cross the line, they probably do, the same as paparazzi do, though probably far less often since their entire job is expensive to pay for, and they won’t be paid if they end up in trouble for breaking laws while doing something unnecessary.
In other words, movies and TV vastly misconstrue what the job is actually like.