The two bodies will actually revolve each other in an ellipse-shaped orbit.
The only "real" force that applies to these two masses is actually just the a=Gm/r2 gravity acceleration. The centripetal force is apparent, like an illusion, but does not actually give any real acceleration to the objects.
So the two bodies will keep on orbiting each other, because the gravity attraction that pulls them directly towards each other also have to comingle with the already existing vector motions between the two objects.
For example, if the Earth were to be made suddenly completely still with respect to the sun, then our planet will immediately be pulled in straight line towards the sun and within days the Earth will collide with the sun!
So it is only because the Earth is already moving with respect to the sun from time immemorial that, no matter how long the sun keeps pulling us, the end result is that our two masses will just revolve around each other indefinitely in an ellipse-shaped orbit.
The "center of mass of the two body system" is more like an added analytical concept to this, so we might choose use the concept or not depending whether we feel it is helpful to aid our thinking.
If we choose to use the center of mass framework, then we can start to derive mathematical relations etc. about how does the center of mass will behave with respect to the members,
how the center of mass will shift over time as they rotate (or not), etc...
Hope it helps!