In Canadian English, to be pissed can be both to be angry or to be drunk.
And then there is another expression to explain that someone has no money.................. "He doesn't have a pot to piss in or a window to throw it out!" Not a very nice expression, though.
Not business, but they'd pass the 'mum test' suggested earlier in my case. ;-)
Perhaps a better test is the "new fiancee's mum test". This has the same flaws, but in this encounter you are in uncharted waters.
Attitudes toward the vulgarity of a given phrase or word can change.
Piss was one of George Carlin's "Seven Words You Can Never Say on Television" back in 1971 or so. This would have included any of the variants I can think of (piss=urinate; piss off=anger, go away; pissing contest=pointless argument; piss away=waste; piss on=ignore, insult).
Cut to the late Nineties, early 2Ks. I can't tell you exactly which episode of "The West Wing" it happened in nor can I quote the exact lines, but I can remember being somewhat jolted when I heard First Lady Abigail Bartlet, in a conversation with her husband, saying someone had pissed someone else off. I have heard it more than a few times since then.
This is not to say piss in other contexts is acceptable on TV, though other analogies are so (e.g., "take a leak"). Personally, I still consider "piss" vulgar, regardless of the context. I will occasionally describe something as a "pissing contest," but that's about it.
So, basically, thirty-some years later, piss as it appears in to "piss off" is now OK to use on TV, and we're down to "Six Words You Can Never Say on Television."
When the critic Kenneth Tynan first used the f-word on the BBC in the 1960s, his stammer led to the joke that this was the first 13-syllable 4-letter word. Sadly, the BBC didn't keep a copy of the programme.
Still, I think the thread needs to stress the difference between the degree of vulgarity in the imperative "Piss off!" and the verb meaning to anger. The former is Hell's Kitchen nastiness and the latter is only mildly vulgar, in general.
I would not use the word in front of my children. Instead of "Piss off," I would say, "Beat it," or "Get lost," and instead of saying I was, "Pissed off," I would say I was "angry" or "mad." That's just my opinion, but remember, just because something is on television or especially the Internet, does not make it appropriate. :-D