There are a few things to consider here. First, almost by definition, your opponent circling you will be expending more effort than you if you are staying in place. Imagine that you're at the hub of a wheel and your opponent is at the rim. You will be traveling a shorter distance to turn a given arc than the opponent. This, of course, may vary if you, in turn, circle them.
Second, you will want to be careful about how you step to avoid tangling yourself up. Obviously, most trained people will not try to just pivot in place and wind up crossing their ankles so that both legs can be swept at once, but it's not terribly uncommon for people to put themselves in an unstable position for "a moment" before correcting their balance. It seems harmless, but if you are attacked in that moment of unbalance, you are much more likely to react in a way that will keep you unstable.
Third, you will want to avoid being too predictable in how you move or else your opponent will learn that when they circle you in a particular direction and speed, they can predict where your feet and balance will be so that they can throw an attack when you are unstable or at the least in a particular stance. This kind of follows from the advice above about not tangling your feet up or putting yourself in a position of unstable balance, but overall, it's more of a general bit of advice to introduce variation in your movements. Honestly, this also kind of fits into that you may not always want to turn to face your opponent. Sometimes, being a bit sideways will help you to provide a smaller profile as they attack, or lead into using momentum or twisting your body as a spring to drive an attack.
Lastly, be aware of your environment, particularly what's behind you that you can't see immediately. In a ring, an opponent may circle you to trick you into backing into a corner, the ropes, the fence, or the boundary that indicates a ring out. On the streets, they might try to maneuver you onto more unstable ground, a solid surface, or even a drop, which will reduce your movement options.
So, what should you train?
Fundamentally, you will want to practice a few different movements so as to not be too predictable. You will want to try to find a movement that keeps your feet stable at all times. You will want to practice the movements until they feels so natural that it's virtually effortless. Personally, I like to try to lift my feet a fraction of an inch above the ground and trail a small part of my foot on the floor so that I'm not wasting effort with friction by sliding my feet, it's easy to set my foot down, and if I run into an obstacle, I know immediately and can deal with it. My default involves sliding the foot opposite the direction they're circling back, and then the other foot forward because it keeps my legs more apart, but I'll generally mix that up with stepping other directions or doing a quick shuffle. In the latter case, I won't always turn in place, or circle their direction, but will sometimes instead step into their arc to create a threat.
