According to Reddit user and known Destiny leaker AnonTheNine, development on Destiny 3 is "starting right now," and the sequel will double down on the "hardcore" RPG elements which Bungie initially shied away from in Destiny 2. AnonTheNine also said Destiny 3 will be helmed by Bungie game director Chris Barrett, adding that Destiny 2 director Luke Smith will still hold "a big role."
Traces of Destiny 3 started appearing earlier this week when AnonTheNine posted the above image of Europa - one of Jupiter's moons which, as our sister site PC Gamer reported, has appeared in Destiny concept art in the past. It seems Europa will be a key area in Destiny 3, though AnonTheNine didn't specify whether it will be some sort of hub or one of the game's (presumably numerous) patrol areas.
More recently, when Reddit user AntimatterAccord posted about Bungie's refusal to attach clear percentages to the effects of armor and weapons in Destiny 2, AnonTheNine chimed in to say that Destiny 3 will deliver the numbers and stats that diehard players are hungry for.
"D2 vanilla was their idea from [the] beginning," AnonTheNine said in a comment. "The 'reboot' didn't change that, the game was born to be for casuals. If D3 will really have the idea they want to implement, trust me, a lot of Guardians who play two hours per week will abandon the game. They are going balls out with the RPG stuff."
In another comment, AnonTheNine said Destiny 3 "will be even more 'hardcore-y' than Forsaken" and will more strongly "push the RPG side of the game." They also described open-world areas where players can kill each other, as well as Guardian powers fueled by the Darkness - the elusive antagonistic force behind Destiny's universe - rather than the Traveller's Light.
AnonTheNine previously predicted new subclasses and areas in Destiny 2's Forsaken DLC, and more recently the return of Thunderlord in the ongoing Festival of the Lost Halloween event, so they do command some credibility as a source. That being said, none of this information has been verified, so all of it should be taken with a grain of salt. I reckon we won't see Destiny 3 for at least two years, and plenty can change in that time.
Destiny 2: Forsaken is the best Destiny has ever been. If Destiny 3 can improve on it, it'll be something to see.
