On Xbox, you can play those games, whether signed into xbox live or not, on the original system you purchased them on.
If you want to play those games on a system other than the one you originally purchased them on, you either need to be signed into xbox live on the account you purchased the content with, or transfer the license to the digital content over to the new system to be able to play the games on that new system without being signed into xbox live.
Once xbox live goes down for xbox 360 in a few more years, you will only be able to access your digital content by playing on the original system you purchased it through (or whichever system you transferred the content license to). When that system dies, those games are gone.
I steer clear of digital content for this reason. If Sony or MS allowed you to transfer or redownload that digital content onto the new gen systems, I'd feel differently, but they don't. So I don't spend money on digital games at all.