Entirely upto the agent. I have had UA agents who'd not even check-in bags to connecting UA flights on separate tickets, and I've had agents who'd do it on separate PNR/tickets/carriers. If you are flying out of SFO, I think you have a good chance (the agents there, specially in the premier lanes know how to handle external PNRs), if coming from an outstation, good luck.
It is true that in many Asian Airports (SIN, BKK, KUL, HKG etc). the airlines will do a transfer if you give them your bag claim at the transfer desk when you get your boarding pass, but you do need extra time - as they literally go and find the bag from the baggage claim and re-check/re-tag it. But if your baggage allowance differs, then I am not sure how it gets handled. the SIN-MAA segment may not have the same bag allowance as USA-SIN.