Who would go to them to to offer terms or treaties? Would they usually go with a show of force like the backing of a Legion or a smaller diplomatic group? Would this always be under the direct authority of the Senate or Emperor, or would a local governor or military general be able to authorise such action? I would be interested in examples.
If you're asking for contact during the Roman Republic, it was really anything goes. How the contact went largely depended on how big of a threat they were to Rome or its peripheries. Gallic tribes were militarily a huge threat to Rome and its colonies along the border of Italy, so there was a larger incentive for war rather than diplomacy. But for just general authority of wars, usually the senate as well the tribunes and assemblies would decide whether or not to go to war, and the advice of the senate was generally followed. From that point the elected Consuls were free to wage the war. A simple google search for the political structure of the Roman Republic will probably do a much better job at explaining this than I can!
One huge example is actually Caesar during the Gallic Wars that lasted from ~58-50 bce. For this, the military was taken from Rome as well as local "bases" in the area. He gained a lot of prominence for conquering Gaul, which boosted his political career. In fact, you can read his personal account of those wars in Commentarii de Bello Gallico!
Main Source: Boatwright, Mary Taliaferro., Daniel J. Gargola, Richard J. A. Talbert, and Mary Taliaferro. Boatwright. A Brief History of the Romans. New York: Oxford UP, 2006. Print.