Past act 2 Republic and Empire basically re-entered full on active war, the Battle of Corellia was massive, costing the Empire 10% of it's military force. Then they started a Battle for Ilum but that was cut short by one Darth Malgus. And throughout the years, they continued. When you do the Makeb Imperial story, Marr informs you that he lost 3 systems trying to bide time for your operations on Makeb. The Republic and Empire invade Korriban and Tython, respectively. Over Rishi, the Imperial and Republic fleets start a massive battle that only ends after your interference. The team-up on Yavin isn't Empire + Republic, it's Marr's people and Satele's people specifically.
It's just that the player characters take part in only select conflicts and completely miss out on other things that may happen throughout the galaxy.