Why do I see a green dot only occasionally next to some saved messages but not next to other saved messages?

The green dot indicates an Unread message. You can see this in action by repeatedly pressing the "M" key, which toggles it between Read and Unread status. Only you can can prevent others from accessing your computer and Thunderbird. Nobody here has any way to get to it. You might consider moving any sensitive messages to a local folder, then deleting them off the mail server in case the provider is compelled to give an authority access to your mailbox. But that might not help, either, if the provider has backups, which they probably do.,