This is something that the LFL Story Group has always played pretty damn fast and loose with; re: Sith, and it's biting them in the ass.
A graph plotting Palpatine's "at-risk" status probably looks like a bell curve, with different time periods having different risks:
Stage 1: Palpatine the Apprentice
At this point, Palps can freely move around and do dirty deeds -- he's Plagueis' hatchet man at this point -- going around and force choking people who are loose ends. His risk is fairly low -- he hasn't done anything (yet) to rate him high on the Judicial Corps or the Jedi lists of people to look at.
The mysteriousness of the Sith "extinct for a thousand years" plays into his safety here -- the Jedi won't dig too deeply since lol fairy tales. But IF they do think a Sith is really walking around; they're going to squeeze Palps for info to find out who the Sith is. Hence he's at high risk of being liquidated by Plagueis to prevent loose ends, so on one hand, on the other hand etc.
Stage 2: Palpatine the Sith Lord
The Tragedy of Darth Plagueis the Wise has occurred, etc. Now Palpatine is Big Boss. Presumably around this time is when he begins his slow rise through the political ranks. It could be that in the new canon the Plagueis/Sidious breakup was a disagreement over methods to achieve the Revenge of the Sith -- Plagueis wanted a traditional "lightsabers at dawn dueling over Coruscant skyline as our troops burn the city" thing, while Palpatine wanted a primitive version of what became Order 66.
Now that he's insulated from direct murders -- he gets other people to do those now -- Palpatine's risk drops, but as he moves up in the political ranks, his risk rises. Why? The average prole isn't going to have a lot of people investigating them, but an upcoming Vice Legate is going to have their history dug into by all sorts of people seeking political dirt.
Stage 3: Palpatine the Senator
It took a lot to become Senator of Chommel Sector (even though it's rather sparsely populated by Republic Standards); so Palpatine is still kind of in debt to his political patrons. Even more people are watching him now, from blackmail gangs (can you imagine the fun he would have had with a Blackmail gang that tried to blackmail him over certain sudden and unexpected deaths?) to other Senators looking to take him down a notch.
Additionally, the Jedi are now watching him somewhat, so he's got to be better at his "force shrouding" skill.
Stage 4: Palpatine the Chancellor
Now that he's Chancellor, he's gained some natural immunity to certain vectors of attack -- "You expect me to believe that Chancellor Palpatine is responsible for five dozen murders? And not once during his rise did anyone find out about them? Not the Royal Naboo Constabulary, the Republic Judicial Corps, or the Jedi?" as pointed out earlier up thread by Lord Revan.
But he's also quite vulnerable -- he has to expend a lot of effort now in using Force Shroud and controlling his emotions now that he has full time Jedi minders around him and he's in direct near-daily contact with the Jedi Council; rather than the low ranking Padawans he had contact with earlier as a Vice Legate, etc.
Stage 5: Palpatine the Galactic Emperor (Early Years)
I believe this is where Palpatine's personal danger quotient peaks. Yes, the Jedi are exinct, their survivors driven into hiding in the swamps of Dagobah or the sands of Tatooine; but politically, his regime is quite weak.
That's always the danger with revolutions -- once you light that tinderbox, it's easy for a counter-revolution or another group of revolutionaries to knock you off your pedestal.
But the main reason is that what was once incredible is now...everyday.
The Jedi trying to coup the Republic?
If that's possible, then other things are possible as well -- i.e. he's now more vulnerable to leaks that prove he did horrible thing x y or z while a small politician on Naboo.
And above that, he has to keep the truth about the final days of the Republic tightly sealed; because the legimitacy of his entire regime rests on the people accepting the public story.
Stage 5: Palpatine the Galactic Emperor (Late Years)
At around fifteen years into his reign, he's in a lot less danger -- everyone who could possibly pose a threat has been carefully neutralized or eliminated and almost an entire generation has grown up with Emperor Palpatine from their first memory.
I believe this is where he started bringing the Sith "out of the shadows", so to speak, with actual Sith Cultists studying semi-hidden on Byss -- i.e. they practice Sith abominations in the open on Byss, without the need to hide it behind euphemisms and ceremonies at midnight; but to the known galaxy, Byss doesn't exist on star maps.
This might be why Vader is so nonchalant about namedropping his line about destroying a planet being insignificant next to the Force on the Death Star in ANH; he's been to Byss and seen what they've been up to.
Stage 6: Palpatine the Sith Emperor (Unrealized)
As mentioned earlier upthread by me -- eventually Palpatine wanted to go "public" with the Sith being the overt rulers of the Empire from the new capital of Byss.
At this point the only thing he'd have to fear is old age or assassination attempts, both of which can be averted through the powers of the Sith; at which point he's Sith-Emperor Palpatine resting on the Durasteel Throne, a thousand force sensitives sacrificed to him every day, etc etc in the Grim Darkness of the Future there is only war...