The Star Wars comics explain why Obi-Wan Kenobi was never found by the Empire on Tatooine. Palpatine's triumph in Star Wars: Episode III - Revenge of the Sith left the surviving Jedi reeling. Only Yoda and Obi-Wan Kenobi knew Anakin Skywalker had become Darth Vader - but they believed there was still hope. Anakin's beloved wife Padmé had given birth to twins before she died, and the two Jedi believed the infants had inherited the potential of the Chosen One. Senator Bail Organa of Alderaan adopted Leia, while Obi-Wan took Luke to Tatooine, where he retired to watch over the boy.
Despite the tragedy of Obi-Wan's mission, the decision to take Luke to Tatooine was an amusing one. Obi-Wan knew Darth Vader would never willingly go to Tatooine, because the desert world held too many bad memories for him. His logic largely appears to have been sound, but the success of his hideaway has often been challenged as unlikely, especially given the Dark Lord of the Sith established a group of Inquisitors - Force-sensitives whose job was to scour the galaxy looking for Order 66 survivors. Obi-Wan didn't exactly live a quiet life once he got to Tatooine - he's known to have caused quite a bit of trouble for Jabba the Hutt - so it's surprising he was able to remain in hiding.
The opening arc of Jason Aaron and John Cassaday's Star Wars, which launched in 2015, explains why the Empire didn't learn Obi-Wan was on Tatooine. It seems Jabba the Hutt's criminal empire flourished during the Dark Times of the Empire's reign, even earning him the title of Warlord of the Outer Rim. But Jabba understood Palpatine's Empire was greater than his own, and he was wary of drawing its attention. Ironically, this ensured Jabba had good reason to keep a lid on any rumors of a Jedi Master on Tatooine.
The destruction of the first Death Star, accompanied by Rebel attacks that disrupted the Empire's supply chains, later forced the Empire to seek assistance from Jabba the Hutt. Darth Vader himself was sent to Tatooine, and by now he'd figured out Obi-Wan Kenobi had been hiding there for years. In Star Wars #24 he took the opportunity to quiz Jabba on Obi-Wan's activities, attempting to find out where the boy who destroyed the Death Star had come from. The scene is an entertaining one, with the Dark Lord clearly seething to be back on Tatooine - all the more so because he realizes his old master was able to take advantage of his personal weakness by hiding on his own homeworld.
This does raise an amusing possibility: that there were other Order 66 survivors who successfully hid on Outer Rim worlds, and who took advantage of the locals' fear of drawing the attention of the Empire. It will be interesting to see whether or not this idea is developed in the upcoming Obi-Wan Kenobi Disney+ TV show, the first Star Wars live-action TV series set during the Dark Times. Whether or not that's the case, it's an amazing narrative decision to have Obi-Wan's ability to hide from the Empire on Tatooine depend on the competing evils of Jabba and the Emperor. While Obi-Wan made a smart decision hiding from Vader in a place he'd never want to revisit, Star Wars also makes it clear that it was the lust for power of his enemies that truly kept him safe.