@Stolas has already explained that the only way to be sure what an application does is to reverse engineer it and inspect its code, and @RoryAlsop already described why such access permissions are required from the application architectural point of view. But there's one thing that I feel I should add.
I think there's not much to worry about here. Why? LinkedIn is a fairly big player and as such under constant scrutiny of the public eye, like all the big ones are. If they were up to no good and trying to access data you didn't agree to in their TOS, and/or otherwise misuse them, they would have to deal with big problems keeping that under the rug and risk huge loss in their reputation and credibility, possibly even be a subject to legal prosecution and financial loss that would come with it, if it were ever to become public knowledge.
You see, these apps aren't developed by a few tightly controlled developers kept in some basement and only allowed access to daylight once thoroughly brainwashed for any residual disclosing information. I'm being slightly sarcastic here, but I believe that living under constant paranoia is even more damaging to one's mental health than my opinion compressed in a few lines could ever be. Anyway, if LinkedIn (and this goes for any other big player in the field of social networking out there) was misusing your personal information in a way that is not clearly described in end user agreement (or other such documentation) you agreed to upon signing up for their services, and/or installing their software, chances are extremely big you'd be reading about that in the news and LinkedIn wouldn't exist anymore;
- One of the developers would suffer guilty consciousness and blow a whistle on them to relieve the pressure and hopefully sleep better. Or,
- an independent researcher would find interesting inner workings of the code he/she just reverse-engineered from a signed install package LinkedIn is publishing. Or,
- a sleepless networking expert (not to be confused with script kiddies) would find some such indicative network packets being exchanged between his test client that he setup and a LinkedIn server, that the downloaded app was responsible for. Or,
- an IT security professional will be asked to assess potential threats some company faces with their BYOD policy. Vulnerability assessment will include some of the most common Android device software, and the mentioned LinkedIn Android app will be most likely among the first ones tests will be conducted on.
Regardless who would be the first to discover it, LinkedIn could either be blackmailed and settle it privately (which could still leak eventually), or have to defend themselves in front of the eyes of the public. Both of which would incur cost to the corporation, something they don't appreciate, not in the least bit. And since alternatives to illegally exploiting your personal data are a lot cheaper, that's what they do. They test their code thoroughly for compliance with all kinds of regulations, sign them with certificates that prevent install package tampering, and they're proud to display that to end users too. The rest is then between you (your free will to disclose your personal information to whomever you want), and LinkedIn (the ones that will gladly take it and turn it into profit). This said, it's up to you to decide, how intrusive you'd find such social networking symbiosis, and if you should call it spyware.