Key rules
Russia
The right is set out in Art. 13 of Federal Law No. 115-FZ; you are hired on a certificate from the educational institution, and the employer notifies the Ministry of Internal Affairs (МВД, MVD). For students from EAEU countries the general EAEU rules apply. In some regions foreigners are banned from certain jobs (for example, taxi driving and delivery in St Petersburg). The rules change, so check the MVD website and Gosuslugi (Госуслуги).
Sources: Federal Law No. 115-FZ «О правовом положении иностранных граждан» (On the Legal Status of Foreign Citizens), ConsultantPlus, Ministry of Internal Affairs of Russia («МВД России», мвд.рф): migration services