Actually, there is a standard, for depth, in Open Water Training. I cannot speak for other agencies.
But the actual "dives" versus the training is the clincher. In the SSI environment, we do the skill sets in 30 feet, or less, of water. The first two training dives are between 15 feet and 40 feet (done after majority of academics and pool skills passed), and the last dives in 15 to 60 feet of water, after all academic and pool skills done (and presented in open water).
But in terms of "needing" a DM? Well, they are handy if you're in a new enviroment, need to know where to go, etc. IMHO, I have seen a lot more stuff with a DM or guide along.
But at some point, you should feel really, really comfortable about entirely planning and executing a dive without someone checking on you. It never, ever hurts to ask for advice, however.
Be safe, and have fun!!!