They can be, but many (most?) aren't.
As noted above, until you get your license, it's best to stay at one place.
There are a variety of reasons for this, many are based on 'continuity' and 'familiarity'.
Most DZs aren't super comfortable with someone 'walking in off the street', claiming to be a student, wanting to continue where they were at the previous DZ. At minimum, you'd have to repeat the last level you completed so they could evaluate where you really are vs where you say you are.
But once you are licensed, you can go anywhere you want.
There are some places where local 'rivalry' is present. But, unless the DZO is a dick (and some are), the idea of jumpers hopping around to different places is usually fine.
I've only jumped at 7 different places, but I've never been put down because I jumped at somewhere else. In fact, the opposite is often true. I'm invited to come back again.
Most DZs (and DZOs) don't consider other decent, well run DZs to be 'competition.
For example, Skydive Chicago & Skydive Midwest both draw from the Chicago market. But a couple years ago, the DZO of SDMW was at SDC Summerfest. He sat 'on the couch' with Rook (DZO of SDC) during the filmfest and discussed this. His wife was doing Load Organizing at Summerfest and Rook commented it was nice that he 'let' her do that. First off, he didn't 'let' her do anything. Second, he knows full well that we need to support each other and that will make the sport a better overall place.
I know of a bunch of other instances where area DZs cooperated in a variety of ways.
OTOH, there can be a fair amount of 'cliquisheness' on the DZ itself. Not necessarily in a 'look down' manner, but simply that belly fliers hang togther, as do free fliers, wingsuiters, canopy pilots, ect. They don't really go out of their way to exclude people, but if you aren't 'one of them', they may not be super welcoming.