Some do it before to soften the bristles and make sure the toothpaste sticks to the brush. Dentists say if you do this, then you should just get a softer bristle brush.

Some do it after to make sure it lathers and foams more?

Some do both. Some do neither. Which one is right?

According to dentist...it doesn't matter as long as you're brushing your teeth. And doing it correctly.

Some do say that wetting the toothbrush at all can dilute the toothpaste thus making it not as effective. If you must use water at all on your brush either before or after, keep it to a minimum. And they suggest you don't rinse out your mouth after you brush either. That means no mouthwash after brushing – it could do more harm than good, and it’s much better to use mouthwash midway through the day instead – and no holding your toothbrush under the tap for longer than a second.(Metro UK)