This summer, 2015, I will be travelling to Russia from the United States. A Russian friend invited me but now he will not be able to be in Russia during our trip. We are considering staying at private apartments through Airbnb, which would require our host to register us. However, this host would obviously not be the person who invited us.
Will we have any trouble attempting to register a private visa with someone other than the host on our invitation?
As someone who has undergone the registration process many times (including as recently as the end of 2014), I can tell you that the "invitation letter" (or anything to do with your invitation) is completely irrelevant to the registration process - it is NOT on the list of documents required to register you as a foreigner. The person registering you can be anyone (but they do have to have proof of ownership of the property where you're being registered).
As stated here -- How can I get an 'invitation' to Russia? -- the invitation letter is more like tradition rather than, as Eugene stated, a required document to register.
From personal experience, the invitation letter is really more of a subtle 'tax' to add on to your typical visa application; many 'invitations' can be bought for a small fee, even if you don't stay in that particular hotel/hostel/room.
