[Janus]

If an observer has a velocity relative to you then you will always agree on that velocity because it's the same for both of you.
So if they were to accelerate for time in the same direction as the relative motion from your perspective so that they're now steadily moving away from you at say 0.8c, then from their perspective you are moving away from them at 0.8c despite the fact that they are the ones who accelerated from both perspectives.
During that acceleration the velocity is increasing for both observers . From your perspective their velocity moving away from you is increasing due to their acceleration, but they see your velocity moving away from them increasing, because it's the velocity that you're moving away from each other.
All worked out without once considering which one was actually the one moving and which was at rest.

There is a difference between an inertial frame and an non-inertial one.  You can tell if it is you or the other observer which is accelerating.  If you place two clocks in each frame separated in a lie parallel to the acceleration.  The observer in the non-accelerating frame would note that both of his clocks would keep time with each other.  The observer in the accelerating frame would note that his tow clocks will not; the clock in the direction of acceleration will run faster.
The two observers will also not differences in how they measure what happening to each other's clocks.  The inertial frame observer would simply note that the other observer's clock runs slow by a rate that only relies on their relative motion with respect to him.    The non-inertial accelerating observer will note an additional factor effecting the other observer's clock. This additional factor depends on the Proper magnitude of his  own acceleration and the distance and direction of the other observers clock.    If it is in the direction of the acceleration, this will tend to make the clock run fast and if in the other direction slow.

Also, concerning your earlier claim about a lone object in an otherwise empty universe not being able to measure its own rotation/acceleration.
While it is true that Einstein in his original formulation of General Relativity accepted Mach's principle,  He later reevaluated, that position and abandoned it. He concluded that an object in an otherwise empty universe could  and would detect its own acceleration/rotation.