I think when Eddie is out of town, which happens with a degree of frequency given that he's a musician, Steve is sad, but fine. He has a routine. Running every morning, work, going to the local park and cafe on his off days.
He misses Eddie like a limb, but he knows how to be on his own.
If, for any reason, Steve has to be gone while Eddie is home, it's another story. Eddie is a feral, wilted chihuahua. He's showing up at everyone's house, lamenting about being left alone. He sleeps on the couch just to be dramatic. He sends Steve postcards every day, even if most of the trips are too short for any of them to actually reach him.
Steve has learned to brace himself when he opens the door, ready for Eddie to leap into his arms and, more than likely, lick his face.