Belarus-Russia border crossing should be fully legal as of 2020-07-01
As of 2020-07-01, Belarus and Russia reportedly recognise each others visas:
Foreign Ministers Vladimir Makei and Sergey Lavrov signed the agreement on mutual recognition of visas between Belarus and Russia in Minsk on Friday, 19 June.
Belarusian Foreign Minister Vladimir Makei explained that the document provides for full mutual recognition of national visas and other documents related to the entry and stay of foreign nationals.
“Now foreign citizens have the opportunity to visit both countries with the national visa of one of them. Thus, the agreement removes the problems of third-country nationals crossing the Belarusian-Russian land border,” said Makei.
Under the agreement, foreign citizens with a Belarusian visa will be able to freely enter Russia and, vice versa, holders of Russian visas will be allowed to enter Belarus. Foreign visitors will reportedly be able to travel by air, rail and car.
Source: Belarus And Russia Sign Agreement On Mutual Visa Recognition, Belarus Feed, accessed 2020-07-05. Also reported by the Belarusian Telegraph Agency, TASS, and intourist.
Warning as of 2020-07-07: another source in Russian, dated 2020-07-01, states it will take several more months until it actually takes force. These sources contradict each other. Either it's already in force, or it will be soon (and as of July 2020, the 2019-2021 Coronavirus Pandemic means few travellers will make this trip anyway).
2019-2021 Coronavirus pandemic note: Russia and Belarus may still have different COVID-19 entry restrictions, but this is independent of the new visa regime. At the time of writing (2020-07-05), most foreigners are banned from entering Russia but allowed to enter Belarus, and the Russian-Belarusian border may still be closed to everybody. Now is not a good time to travel as a tourist to either country, but the new visa regime is good news for international travellers once the situation normalises.