I may be restating the obvious . . . my apologies if I am.
Messages on the Mac is dependent on YOU signing in with your AppleID/iCloudID. Two discriptions that mean the same thing. In other words, you Apple Account. Once in, you can send text messages to any other Apple user who has signed in with THEIR AppleID/iCloudID. Your transmissions will appear in a BLUE bubble if you're sending to another Apple account that is signed into THEIR account. If it goes in GREEN, then the receiving party is not signed into THEIR account. With out the added benefit of an iPhone, you cannot send text to a non-apple user from your Mac using Messages. To send to those folk, pick up your iPhone and tap away. Have I missed anything?