By reading the Wookieepedia article on Tatooine, it's seen that the planet undergoes numerous changes in leadership. I can't, however, find any reference to slavery being outlawed.
We know from The Phantom Menace that the Republic at the time had anti-slavery laws, but Tatooine was not under the control of the Republic. It later became
under the control of allied with the Galactic Empire, but seeing as they're the badies, I wouldn't be surprised if they were OK with slavery.
Anakin mentions in Episode I that he dreams about one day becoming a Jedi and freeing all the slaves. Unfortunately, he never gets to do this.
Did it ever happen, or did slavery continue on Tatooine?