People in Photos

There are two places, where Photos could be showing this message.

1. Initially, you will just be seeing this message when you open the main People album. Then you should also be seeing a count down with the number of photos that still need scanning for faces. This can take a really long time because it requires an enormous amount of processing. For my moderately sized library with just 50000 photos and videos it took roughly two weeks.
2. When you open albums for individual contacts and are seeing a message about "PEOPLE will finish updating when PHOTOS is in the background" you can usually safely ignore it. There are occasionally some tagged people in your library, where Photos is not able to get started with this particular person, because there are not enough photos of this person in our library. I have a few people albums for persons, that I will never meet again to to be able to take a second photo. I have only one scanned family photo of them, but Photos needs at least a second photo to be able to learn what the person looks like. These particular albums will be showing the message forever "PEOPLE will finish updating when PHOTOS is in the background". If you have more photos of the person in question, try to find them and check, if Photos has marked the face with a circle. Then add a name to help Photos to get started with this person.

An example, what it looks like, when there is only one face in the album of a person:

It means photos is still scanning your library looking for faces to be named. It can take a long time, and will only do it while photos is open, but not being used (so it is in the background)

Best thing to do is just have photos open as long as your mac is on, and just get on with other stuff - or just leave it on.