I've worked as a recruiter for 7 years, and I have sent LinkedIn print-outs to clients several times. A strong candidate's a strong candidate.
> Recruiters take your CV, remove identifiable information (to avoid company going directly after you), add 1-2 cover/intro pages and send it to the company.
Recruiters have terms in place with their clients that mean the client would have to pay either way. The only time recruiters will anonymise CVs before submission are if they're using a CV to try and tempt a company who isn't a client yet -- which is illegal unless they've cleared it with the candidate first.