According to... etymology...
Ninja... the literal definition of it is Spy.
As in... Ninjas aren't strictly assassins, in fact that isn't their primary job.
Anyhow are Ninjas Real?
Not the fictional version of course (Obviously they didn't wear black garb...)... but like... organizations of spies or assassins?
Spy is misleading too, it implies a hard focus on intelligence-gathering; I'd say the best translation of the phrase given the individual characters' translations I found is 'covert agent'.
Which is a bit unwieldy, but precise. A covert agent is someone who a) acts on behalf of someone else, b) without announcing to everyone and their dog that whatever they do is on behalf of someone else.
So, a ninja is some guy you hire as, essentially, a private criminal. What they do depends on what you're hiring guys for - could be assassination, could be snooping around, sabotage, theft, whatever.
As such, there usually isn't an organization in the traditional sense, as in you go to a big building with Ninjas-R-Us, say hello to the receptionist, and say you would like one medium-sized ninja please. It's more along the lines of how The Constant Gardener put it, paraphrasing: 'You call your Chief of Security, he calls one of his old friends, he calls some people, they call people...' and then the next thing you know some asshat is stabbing your lower intestine.
You could train some operatives in-house - that obviously requires resources, and potentially can be traced back to you easier, but you have them on retainer, and you get this CIA-esque structure - but you probably don't want to share that.
On the other hand, apparently Japan DID develop more formally organized groups as the stronger mercenary groups established monopolies and some lineages of sneaky villagers up in the mountains developed traditions of training more sneaky villagers who sneak gud, but it was never a centralized political entity.