I have no problem with people knowing my real name... And in fact, I have insisted a realname policy on many systems I have moderated during this millennium.
But beyond the real name, it gets complicated. I know there are many people who know my address and phone number that probably should not know it (because it was written in a very public international registry for years; not sure how many did bother to look it up though). But should a need arise, I can change both at relatively little hassle.
So far, there has not been any problem.
But what are the risks, really?
I have no problem if someone who used to know me only from the Internet will suddenly call me, or send me a letter, or even appear behind the door, as long as they aren't stalking and harassing me, or posting that information in some services that send you eh, spam, through those channels. Most people I know though, are either so introverted, or just unwilling to spend money in any of those communication methods, that none of that happens.
By principle, I never permit giving any such information for marketing purposes, if at all possible.
How about my bank account number? So far I haven't been able to see any problem with having it public. To my knowledge, there's nothing you can do with knowing someone's bank account number except send them money, and I have no qualms with that.
I suppose the bigger problem is with the personal identification number (or social security number). Many services use it for identification and authentication of the person. For example, if you pay by a bank or credit card in Finnish stores, and your card does not have the smart chip in it, they take your signature, and beyond a certain sum, write down your SSN. They also take your SSN and signature in the post office when you fetch a package. In addition, if you travel by train without a valid ticket and get caught, they ask (and verify) your SSN and use that for inspection fee related purposes. Technically, you could give someone else's credentials in all of those situations, but it would be a legal fraud.
There may be some people who know my SSN that should not know it, but in general I have avoided typing it anywhere in the Internet. Also a few people have seen my signature, but there's little value in that. I could scribble anything, really.
There are only a few situations where I truly wish to be completely anonymous, and I realize it is quite difficult really, especially in the Internet. I am not used to using proxies like Tor, and in general I do a shoddy job at masquerading my own style. I use certain types of sentence structures, and a particular wording at times, and a person who has a suspicion might cross-reference and determine a match.
In a related topic: I was actually surprised to learn yesterday, that in USA it is practically a constitutional right to be anonymous in any situation, even in legal and commercial situations. You could have a pseudonym and use it anywhere, and it would be permissible. Not so in Finland! In fact, in Finland we are basically taught and educated in such manner that if you introduce yourself using a false name somewhere, it FEELS wrong and that you are a terrible cheating bastard for doing so. In Finland, we are taught that in any and all official situations, like opening a bank account, or cashing in a cheque (should you still receive such thing from some undeveloped country), or voting, or subscribing to a magazine, or when a cop pulls you over in traffic, or indeed, picking up parcel in the post office, you will have to prove your identity in an unambiguous manner (usually by showing your papers, which might be the driver's license, or a personal identification card, or a passport). It is a very different world.