If you (or especially a celebrity or well-known person) were "spotted out with someone," it means you were "seen in public with someone." Since you usually hear it used this way in tabloids or sensationalistic journalism, it implies that the people in question didn't wish to be noticed.
"To spot" doesn't always have a negative connotation though. While it's true that you might want to spot your enemies before they spot you, you might also spot your friend in a crowd, spot a great parking space or spot an empty seat on the bus, all of which would probably make you quite happy. When in doubt, substitute the correct form of "see" for "spot," and you'll still be making perfect sense.