When I joined Facebook (back in Jun 2007), I wasn't too impressed when they asked me give them the password to my email account. LinkedIn do exactly the same thing, and I'm not going to tell them either. So, I'm now doing the same thing that I did with LiveJournal, FaceBook, and Twitter: finding one or two people who I know, then "raiding" their contact list to add extra people to my list. So, don't be surprised if you get a request from me soon.
LinkedIn say: "Important: Only invite people you know well and who know you." So, how well do you need to know someone to add them? For instance, I try to keep my list of Facebook friends at about 50 (based on the Monkeysphere theory), but I've seen other people with several hundred "friends", and I'm sure they don't really have meaningful relationships with all those people. One guy used to get mopey whenever his number of friends went down, because he didn't know why people were abandoning him. My view was that if he couldn't actually work out who was missing (i.e. he couldn't remember them) then they weren't really friends in the first place. It's tricky, though: sometimes I've been hesitant to add people (e.g. SJA colleagues) because I don't want to seem pushy, but I also feel guilty about turning down requests from people I don't know particularly well. If I'm "connecting" to people on LinkedIn as colleagues rather than friends, I assume that means that we're not expected to be drinking buddies. On the other hand, at a big company there will be colleagues that you've never even met. So, does it simply mean "Yes, I know who they are, and I agree that we work together" or does it mean "We're on the same team and we talk to each other every day"?
There was an odd glitch earlier, where I'd view someone's list of connections (i.e. "friends of a friend"), but whenever I clicked on someone it said "You and this LinkedIn user don't know anyone in common". That seems to have fixed itself now, but it hasn't given me a great first impression. (The people in question also had missing photos, which have now reappeared.)
Edit: After it fixed itself, it's now broken itself again.
More generally, I still don't really understand what the site is for. I use LiveJournal a lot as a way to pontificate at length (ahem) about whatever's on my mind. I don't post much to Twitter, but it's handy for short messages like "Good episode of Dr Who tonight" or "I've just voted" which don't really merit a full blog post. I mainly use Facebook to plan events and look at people's photos. As far as I can tell, the theory of LinkedIn is to get a foot in the door, e.g. "I'd like to contact person X and we both know Fred so I can ask Fred to introduce us". However, that means that person X is on LinkedIn, so you could just send them a message directly. Is it more of a job-hunting site? If people are posting their CVs etc. then it mainly looks like public info rather than personal data, so I don't think there are many privacy concerns about adding random contacts to your list; in fact, you need to make that info public for people to be able to find you.
Anyway, it's not costing me anything to have a profile (since I won't pay for the "Pro" option), so I'll see what happens. Maybe I'm just missing something obvious here, which will become clearer with time.