This has been awkwardly illustrated. Generally, OsO4 is only used in a catalytic amount. Peroxides are used to oxidize the osmium intermediates back to OsO4. Then after the oxidation is complete, it is as disco suggested, bisulfate removes the peroxides.
I only wanted to clarify this as the poster asked about the bifulfite, but had not illustrated the osmium oxidation with peroxide.