Just like in many European countries, upon arriving in any new dwelling, you must be registered within 7 business days of arriving

Your host at that dwelling (not necessarily the one who issued the invitation) is responsible for registering you. Registration is done at post offices, costs money and involves a lot of hassle.

The proof of registration is a separate piece of paper with a big blue stamp on it. Border guards have neither authority not possibility to check if the duty to register has ever existed and evaded.

Nevertheless, it is worth insisting to be registered at least in the first city you visit. Corrupt check-in staff at dodgy hotels will not let you check in without seeing your prior registration if you've been in Russia for more than 7 business days. Corrupt police and border staff in remote areas will insist that a lack of registration is your fault; it may cost you more than paying the registration fee.

Large hotels are accredited with the Federal Migratory Service and arrange registration automatically and without fee on the day of arrival.