Their actions are known but their faces mostly aren't. You can say the higher ups in the Empire who know Ashe is still alive want to keep it a secret and not make it public or put out wanted posters .. so it doesn't cause riots among the people, same reason why Ashe herself doesn't want to announce her return before she sorts stuff out.
Vaan and Panelo are just random orphans, no one knows who they are. Balthier and Fran are sky pirates so they're good at hiding their identities and Basch, like Ashe, was declared dead so no one's really looking for him.
When they say lay low, they mostly mean not take any actions that will cause them to get questioned etc, like taking the Strahl to Arcadia, the cut-scene when entering Phon Coast shows that the Empire has ramped up air patrol so un-authorized ships will probably not be welcome.
My Ashe is a time and white mage and I can utilize magics for both type at once, so you don't have to change classes each time you want to use a different type of magic.
