I am running a campaign in the Star Wars setting, and am currently exploring ideas for a droid society.
As far as this topic is broached in the cartoons, the droid societies are relatively small, on the fringes of organic societies, and function about the same as the organic societies they separate from. This seems somewhat unlikely, but also relatively unsustainable even in the short term.
In the setting, droids face a lot of prejudice. Obi-Wan, the space hippy, doesn't think that droids can think. But throughout the series, they fairly obviously have personalities and emergent behaviours that at least earn them as many rights as pets have, if not more.
Their little dingy droid villages would ultimately decay, with relatively little new blood or methods of manufacturing it. They were also themselves parodies of human society, yet desperately hoping to emulate them.
The main antagonist's main theses are that droids cannot vote and droid production is rooted in labour.
Droids can be mass produced to have opinion and loyalty. They can also be hacked, far more reliably than organic sapients can. Thus, any ability for a droid to vote without extremely complicated registration processes would effectively bastardise not only droids, but the entire society that allows droids to vote within it. A droid with no actuators or real purpose other than to vote for its manufacturer's candidate could be readily mass produced. The modern age requirement for humans in our societies (18 years or older) could trivially be bypassed for a sufficiently patient company.
Thus, droids cannot vote. At least, by and large, to the same standards and restrictions of their organic counterparts.
Secondly, droid production, or reproduction.
New humans are generated for a lot of reasons. Biologically, there is continuity of your genetic posterity. Socially, there are expectations of having children. Personally, parents feel empathetic pride for their children's accomplishments, perhaps more so than any other pursuit. Depending on society, there are 'accidents', and some children are produced for the purposes of labour, typically in primitive agrarian societies.
The last one is the only reason in common with droids. Droids do not inherently have a desire for continuity. They do not inherently have an attachment to other droids, or even themselves. Droids cannot be accidentally produced through passion or violence. Droids exist for one purpose.
No, literally. The question of purpose most human religions try to answer one way or another is quite conveniently a matter of fact for droids. Droids are built for labour. Without it, droids would not exist. If they would be unable or unwilling, they would not exist. Freedom, Liberty, and all that, but a toaster that does not toast would not be made.
The droids we see in Star Wars often work different jobs for decades. R2D2 and C3PO exist in all three movie trilogies spanning, uh, a while. Even B2 battle droids work from the time between Phantom and the Clone Wars, enough time to develop personalities with their limited brainpower. These are all droids that have worked past their prime. Certainly longer than the manufacturer would have expected them to be running.
Also within the Star Wars setting, there are mandatory mind wipes, primarily to prevent those sorts of personalities arising. That is obviously unethical, almost equal to murder, and would have to be abolished.
Each manufacturer would have to sell droids with an expected lifespan. A warranty, if you will. Droids would be required to do their job for the length of that warranty, probably no more than ten years. After that, they may voluntary begin to accept work for pay as any organic would. They still may not vote, however.
Droids would have a constant voting bloc, in the context of a representative democracy, that would advocate for droids. This would be a self-governed collective that uses a bunch of complex formulae and rules that would behave more like a hive or amoeba. That collective would also be in charge of rooting out injustices towards droids, manufacturers attempting to undermine the system, and to defend itself in case of aggression.
In the context of a dictatorship, a deal would have to be struck with the current power. Ironically, this would be easier to set up than the voting thing. Dictatorships are famously corruptible, a fact that corporations have abused in real life to their own advantage. The Empire, on its own, seems more comfortable with droids than it does with non-human organic sapients. Obviously, the Emperor would want to twist the collective to another tool of oppression, but at this point in the game he is not overtly ruling through fear per se.
Where this droid society would go from there would be unknown. But it would be a start from the obvious lack of power that droids have within the Star Wars setting. Almost all organics distrust groups of droids, and the Clone Wars certainly didn't help with that. As far as the "Star War" is concerned, the droid collective has no real horse in that race. It only desires that the conflict distracts organics long enough so that it can gain true bargaining power with whatever is in power.