It depends very much on the situation and your contract.
Usually, reputable companies who conduct physical pentests have extensive guidelines for their pentesters in many situations. Such instructions are to be followed. I'm going to give a rough overview over possible ways such third-parties may be interacted with:
Local Police
Local Police is to be treated like police is treated anywhere else. A Permission to Attack means that you are legally allowed to conduct a physical pentest, not that you have the authority to ignore the police. Should a police officer instruct you to identify yourself or similar, you have to follow suit.
In fact, the fact that you are conducting a pentest is completely irrelevant to any police interaction. Interact with police in accordance to local laws.
Neighbors
Neighbors are people that have nothing to do with the pentest you are performing. You can't be granted any authority over them, just like the company that hired you to perform the pentest has no authority over them.
That doesn't mean you can't strike up a friendly conversation with them if you happen to see them doing some garden work across the street and see what kind of information you get out of them. But for the purposes of your assignment, they are not any different from regular people you meet on the street. Again, your involvement in this pentest doesn't change anything.
Third-Party Staff
This should be covered either in a general policy of your employer, or specified in the contract of the assignment. In general, they are likely to be treated like any "regular" employee of the customer.
For example, if you can walk inside the perimeter together with the guy who was hired to fix the printer, all the better.
Non-Staff People
Tread carefully! Attempting to social-engineer customers, guests, patients, etc... can very quickly become a very difficult subject for you. This should actually be discussed with the client beforehand, and be explicitly written down.
For example, interacting with patients in a hospital may make them and their families feel disrespected, and you want to avoid that.
Emergency Personnel
In case of an earthquake, a pentest would be the least of my worries. Should emergency personnel appear on-site, aid them as best as you can. You don't know the situation, and somebody's life may be in danger. A pentest can always be done again tomorrow.
Real Criminals
Contact security or law enforcement immediately. Don't play hero, just do your job.