The tackler can and often does pick up the ball. The person that is tackled must release the ball behind them. Once they release the ball they may pick the ball up and get up with it.
Once the tacklee lays down and set the ball they form their "side". The tackler must be on his side before he can return to grab the ball.
I played wing for large university outside of football season - and have also broken this rule 20+ times so I have an understanding of it.
When I tackled another wing in the open field they would lay down, set the ball behind them, while I am jumping over them or going around them to be on side, and then it is a fight to get the ball first. Often they would get the ball again. If I did get the ball in that situation it would often lead to score or big gain. If I didn't get the ball after the tackle, the other wing picks it up and is retackled but at that point there is reinforcement and ruck follows.
And in most cases me or the other wing is going to get cleated and kicked as the guys working the scrum have quite the disdain for wingers. I mention the wing position because this kind of play happens most in the open field where wings would get the ball. Normally in more congested part of the field a player is tackled, set the ball behind them and then a teammate picks it up while stepping over them.