Watching some videos on youtube, it seems that sometimes they use the factory edge as a straight edge (e.g., run it along the table saw fence to cut a parallel side). Other times they trim off that edge by running an edge they cut along the table saw fence. Does it depend on the type or quality of the plywood?
It depends. For the most part yes they are 'straight'. But, more often than not, they are marred and damaged through the course of their transit from manufacturing to you, and will have indentations, of even small fragments protruding from the inner layers.
It is always good practice to start with a clean cut on any store bought wood, and most carpenters, framers, and woodworkers will do this as a standard. If you intend to apply edgebanding, or any finish, or any type of joinery, you should be cutting a new edge.
Now that being said, if you are tight on your yield, the factory edge can be used, depending on the project. For sheathing and framing, the factory edges will always be used for full sheets.
You can check for straight pretty quickly just by eyeing down the edge.
Are factory edges of plywood straight?
But in general you can trust the edges to be straight (and the corners to be square) but never assume they are, check they are.
How straight the edges remain depends on many factors, not least of which are what has happened to the plywood since it was made, including storage conditions, handling and stacking.