My wife and I do this all the time, on vacation, to supplement boat diving. Granted, I am a divemaster, but I did this before I had a professional credential. If you want to do diving on your own (with a buddy of course) you should have sufficient experience to be comfortable in the environment where you will dive. You should acquire information about the depths, currents, temperature, and weather when you will be diving. You should plan your dive, and dive your plan. You should have good navigation skills (at least person in the group) in addition to good diving skills generally. It is important to have a shore support person, or at least someone who knows where and when you are diving, your time of departure and expected time of return. Have a good first aid kit with your shore team, and we always have a repair kit. You should have a diver below flag or buoy on each dive as well. Finally, it is important maintain close proximity to your buddy and to be aware of them and their location at all times. Some of our best dives are on or own, or sometimes with another diver or buddy team. They are made enjoyable by proper planning and safety awareness.
DivemasterDennis