OB/GYN

I would think the disposables would go in the same trash as used gloves, dirty diapers, etc. If the speculum is bloody then it would go in a red bag, like any other bloody trash. I work in a surgical/trauma ICU and we don't even have a biohazard bin in every room -- bloody trash is red bagged, and we carry it to the big red trash can in the dirty utility room.

But it's not a sharp, so unless bloody it's no more hazardous than the rest of the trash.

And I would expect that if there were a child in the room, the parent/caregiver wouldn't allow the child to dig around in the trash

A used speculum does not need to go into a biohazard bin. In fact, in a clinic setting, I fail to see the need for a biohazard bin at all except in your dirty utility room. And a speculum exam in a clinic setting (for Pap smears, STI testing, etc) are typically not sterile procedures, which is the rationale for placing all clean speculums together (although where I've worked, they are processed and wrapped individually, and we keep them that way until use).