InMails are included in paid LinkedIn accounts, both personal and the LIR (LinkedIn Recruiter). I've recently run into InMails working not in the way we'd expect. I sincerely hope that LinkedIn will fix this soon and will update you when they do.
This is what sending an InMail dialog looks like (it has a slightly different layout in LinkedIn Recruiter).
Now, here's what the recipient sees in their email client. (The "subject" and the "inmail text" are created when composing the InMail and contain the specific message to the recipient.)
My contact information, that I had asked to include using the checkbox, when composing the InMail, is not here! I would prefer for my LinkedIn information, such as my location and the tagline, to also be included in the InMail when it arrives in the recipient's email; it's not.
Bottom line: Include your contact info in the body of your InMails. (It will be visible if the recipient looks at your InMail on LinkedIn site, when logged-in, but most people would read these in their email client.) Include your professional info (title, company, location) as well; otherwise the recipient may know too little about you to warm up to look you up - or to respond. Including this information will raise your InMail rate.
This is a short version, with the most important observations, from my blog post.