Yes they did - I've written a little bit about this in the past, so hopefully these answers will be helpful here:

What was the Eastern Roman reaction to Charlemagne being crowned Roman Emperor by the pope in 800? and

How did the Byzantine Empire react to Charlemagne being crowned "Emperor of the Romans"?

And for one occasion when both emperors tried to meet and couldn't because of diplomatic disagreements, see:

In 1185, Holy Roman Emperor Frederick Barbarossa arrived at Constantinople. Do we know anything of any meeting with the reigning Byzantine Emperor, Isaac II?

What did Emperor Frederick Barbarossa think of the Byzantine empire and the Byzantine Emperor as he passed through during the 3rd Crusade? What did the Byzantines think of the Germans?