Basically, there is more than one actual Death Note. Gods of the dead carry them around (they are Shinigami), they each have a Death Note.
When one of Kira's associates (who has a Death Note herself) gets arrested on suspicion of being a fake Kira, the real Kira forces Ryuk to take the associate's Death Note, and Light gives "his" Death Note to the Shinigami that accompanied his associate.
Then, later on in the series, Light uses two associates of his to act as a proxy for his killings. A lawyer uses a fake notebook to act as a decoy, and then he sends the names to another of Kira's associates to actually do the killings, all while Light/Kira pretends to helm the investigation into the identity of Kira.
And giving up ownership of the Death Note causes you to lose all memories of ever owning it, so they wouldn't have to lie while being investigated since they didn't have any recollection of what a Death Note even is.