Legal services for foreign citizens

To register foreign citizens for migration purposes, both the foreign citizen and the host party who will be "registering" the foreigner will be required.
Documents for Registration from a Foreign Citizen
In addition to the arrival notification form, the following documents are required for migration registration:
passport and its copy;
migration card and its copy (except for Belarusian citizens);
visa copy (for foreign citizens requiring a visa).
These documents will be needed to complete the foreign citizen's registration form. To register foreign citizens for migration purposes, both the foreign citizen and the host party who will be "registering" the foreigner will be required.
Documents for Registration from a Foreign Citizen
In addition to the arrival notification form, the following documents are required for migration registration:

A foreign citizen's possession of a patent, timely payment of personal income tax in the form of a fixed advance payment, employment under an employment contract, or performance of work (provision of services) under a civil law contract constitute evidence of the foreign citizen's legal presence in the Russian Federation with the stated purpose of entry as "work." However, foreign citizens temporarily residing in the Russian Federation are also subject to registration at their place of residence.
The patent's validity is considered extended for the period for which the personal income tax in the form of a fixed advance payment was paid. Otherwise, the patent's validity expires on the day following the last day of the period for which the personal income tax in the form of a fixed advance payment was paid. If the previously declared period of stay in the Russian Federation changes (including when obtaining or extending a patent), the place of stay, or other information about a foreign citizen specified in paragraphs 1–9 and 12 of Part 1 of Article 9 of Federal Law No. 109-FZ of 18 July 2006 "On the Migration Registration of Foreign Citizens and Stateless Persons in the Russian Federation," they are registered for the duration of their legal stay in accordance with the procedure established by law.
Registration at the place of stay of a foreign citizen temporarily staying in the Russian Federation is carried out for the period stated in the notification of arrival, but not more than the period established by Federal Law No. 115-FZ "On the Legal Status of Foreign Citizens in the Russian Federation" (the period of legal stay).
To register a foreign citizen at the place of stay, the receiving party may submit the notification of arrival to the territorial body within the timeframes established by Federal Law.

No, it can't.

When registering a foreign citizen and registering them at their place of stay, the host party may be:
Russian citizens who are registered at their place of residence in the country and own residential property at the address where the foreigner's migration registration is being processed.
Foreign citizens who have received a residence permit in the Russian Federation and are registered at their place of residence in the Russian Federation.
A foreigner who owns residential property in the Russian Federation may register themselves. However, they cannot register another foreigner or their family members.
A foreigner with highly qualified specialized personnel status who owns residential property in the Russian Federation may also register themselves. They may also register their family members.
It is also possible to register a legal entity. A Russian company, its branch, or representative office may act as the host if the foreign citizen actually resides at the legal entity's address or in premises leased by the legal entity.
An international organization or its representative office, or a representative office of another country, may also act as the host if they provide residential or other premises for the foreign citizen's actual residence.
A Russian government agency or local government body.
In other words, any of the agencies listed above may act as the host and register the foreign citizen for migration purposes.

The timeframe for registering foreign citizens for migration purposes depends on the foreigner's country of citizenship.

Generally, all foreigners must register for migration purposes at their place of stay no later than 7 days after arrival, but there are exceptions for citizens of EAEU countries and citizens of Tajikistan, Belarus, and Ukraine.

Notification of the arrival of a foreign citizen permanently or temporarily residing in the Russian Federation must be submitted to the territorial office of the Main Directorate for Migration Affairs of the Ministry of Internal Affairs for the Republic of Bashkortostan no later than 7 business days (weekends and holidays are not included) from the date of the foreigner's arrival at their place of stay.

In other words, the host party must submit a completed form to the authorities responsible for migration registration.