When something is expunged or sealed, is it required that that non-government entities purge the records also?

For instance, it's common for local news companies to report arrests & mugshots on their websites. I'm assuming there are similar aggregators for arrests and convictions that span larger areas than a single news company, and these are the same people that service background checks.

What obligation(if any) do they have to respect the fact that a conviction has been expunged or sealed by the courts?

Doing a little research, it appears that you may be right, but, as you say, the requirements vary by jurisdiction. Some jurisdictions will allow you to expunge misdemeanor convictions but not felony convictions.