US SecDef stated that the Russians suffered casualties of up to 350,000 dead and wounded, I find that number quite acceptable.
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Initial invasion force:
100-150 thousand professional soldiers between the SV and VDV
35-50 between DPR/LPR/FSB/Wagner/Chechens and Cossacks
Total = 135-200 thousand
Partial mobilization Oct/22:
300 thousand
60-80 thousand between Wagner's recruitment during the summer of 2022 until early 2023, in addition to volunteers who followed the mobilization of the 300 thousand
Total = 360-380 thousand
Volunteers 2023/2024:
540-640 thousand
Total = 540-640 thousand
Total strength: 1,035,000-1,220,000
Casualties: 350 thousand
Total = 685,000-870,000
Such numbers are very close to what Putin claims and provide basis for what US SecDef also claims about Russian casualties.
Here is a good source for a basis for estimating the size of each side's armies:
Interestingly, both sides may be suffering from a lack of military employment materials to equip their units mobilized and deployed in Ukraine: