Hooboy ... I knew where this was going before I ever read the first response ...
Brianna, the most comprehensive answer I can supply would be ... as in most things related to scuba ... it depends. In theory, your initial Open Water training is supposed to prepare you to dive independent of a dive instructor. But the reality is that it depends on the quality of the instructor, the personality and aptitude of the student, and the business model of the operation who's offering the class.
Most standard agency curriculum and training materials are geared toward the largest diving demographic ... which is the tropical vacation diver. Following this curriculum strictly doesn't really prepare a diver for diving on their own, because it doesn't really teach them some of the fundamental things that an independent diver needs to know. For example, underwater navigation ... the OW class trains a diver to read a compass, and how to follow the compass to do a reciprocal ... "out and back" type dive. But the reality is that nobody ever really dives that way. We go out, wander around a bit, and come back. So the diver is trained in the fundamentals of a skill, but not really trained to apply that skill to real-world conditions. Gas management is taught as a "reactive" skill ... in other words, watch your SPG and end the dive when it reaches a certain point ... rather than as part of your dive plan, as it should be. Even the most basic skill ... buoyancy control ... isn't typically taught adequately, in part because this is a skill that really does take a certain amount of practice to really master ... and for most divers the amount of practice needed exceeds the in-water time allowed for their class. And so most divers end up with varying degrees of discomfort with their skills, and a very limited ability to dive without supervision. For many it isn't their "skills" so much as their confidence level and ability to maintain sufficient awareness of their surroundings that limits them.
It also boils down to environment. Where I dive, visibility can be an issue. OW training doesn't really prepare a diver to dive unsupervised in low-visibility conditions ... even in places where that is the norm. Why? Well, mostly because it takes more than the few dives offered in OW training to really learn how to do it. Diving in low-vis requires combining the skills of navigation, awareness, and buddy communication/positioning. That's in addition to watching your buoyancy control. And it's quite a handful to expect from any newly-certified diver ... most of whom are using up all their mental bandwidth just trying to remember everything that got thrown at them in OW class. In an environnment where the diver can see better there is significantly less task-loading, and a new diver has a much easier time feeling comfortable enough to be able to conduct their own dive independently.
And it boils down to personality. How confident is the diver? How comfortable are they being underwater? These are really individual traits that have nothing to do with the quality of instruction you received. Some people are very self-confident, and adapt very quickly to new things. Others require more time and effort to find a comfort level that would make unsupervised diving a good idea.
While it is a great notion that people should not receive their certification until they have achieved a competence and comfort level that enables them to dive unsupervised ... we all know that's not the reality. One only has to spend a few days at any local or tropical dive site to see people who are certified, but obviously not ready to be planning and executing dives without at least a more experienced diver helping them along. We all would agree that's a level of dependency that isn't desireable in diving ... but we have also seen how common it is.
One can lament that it's because of the "dumbing down" of training ... and it isn't just American training folks, I've seen Asians and Europeans who are just as incompetent in the water as their American counterparts ... but it's really the result of an approach that says "anyone can learn to dive". The fact is that not everyone should dive ... not even among those who put real effort into learning.
Like any other recreational activity, how well adapted you are toward doing it at a given level is going to depend on a combination of your training, your natural abilities, and your motivation ... and these will all depend on the individual. I've seen fresh-out-of-OW divers who look like they were a fish in a former life ... and I've seen people who've been diving for years who still look completely uncomfortable underwater.
So there is no simple answer. There is no simplistic ... "here's how it should be". And unless we want to go back to the bad old days of making scuba diving the exclusive domain of the testosterone crowd, lamenting about the "dumbing down" of scuba education is a waste of breath ... it is what it is for business reasons that have everything to do with making diving accessible to the masses. The best any of us can do ... as dive professionals or just as experienced divers ... is to help our newer members learn the things that they won't learn in a class, and expand on the things that they did learn, so that they can become independent as quickly as possible. And that will be ... for each one of them ... exactly when they decide they're competent and comfortable enough to feel confidence in their ability to plan and execute a dive on their own. It might be right out of OW class, or years and many dives later ... it all depends on the diver.
... Bob (Grateful Diver)