I think the privacy concerns of having your mail (potentially) available over yet another server in exchange for modest convenience makes it unlikely that I would use this, but I'm sure many will find the trade-off acceptable and desirable.
* your local mail client might get different E-mail content every time mail is downloaded, which is not the intent of IMAP,
* LinkedIn (hence, the NSA) gets full access to your E-mail,
* once people get hooked it's easy to transition to inserting ads, or "more helpful LinkedIn content",
I find all this rather disturbing and would never use this service.
What if I believe that Google (hence the NSA) already has access to my Gmail? What's the cost to my privacy if it's already lost?
My major concern is that if I provide Linkedin my credentials, I now have doubled my attack surface for intrusion by non-governmental actors.
Can I have your gmail and password? If not, why not?
You don't have a choice. If the person on the other end is using this service then your emails to them are hoovered.
This is making a big assumption that they understand the implications. Or that LinkedIn explains them at all.