The major concern is with homemade garlic-in-oil. Garlic tends to have botulism bacteria, but they can't thrive in an aerobic environment. When you put it in oil, it will thrive (even when refrigerated) and produce the toxin, which is deadly.

The only way to kill the bacteria is to cook it at an extremely high temperature/pressure, which I believe cannot be created by home cooks (certainly not on the stove).

Store-bought garlic jars are treated to be safe, though I don't know if they can become contaminated when you open them and leave them out on the counter. My guess is this is probably not common, otherwise it would happen with all sorts of food that is stored in oil and kept for long periods of time in the fridge.

Keep it in the fridge.