I work at NIST with this stuff.
Stability and accuracy are often misunderstood. The rate of any clock is always measured against another clock. Suppose you measure the time difference between two clocks over some interval. The frequency is the slope of those data. The STABILITY is measured by how straight the lines are, i.e. how predictable the clocks are.
ACCURACY is more difficult to measure. A clock is first a device that produces a frequency that realizes a principle of physics, e.g. inertia for the earth's motion, or the quantum mechanical energy states of atoms. A clock's accuracy is the extent to which the frequency that a clock produces, matches the theoretical value that it should produce. The inaccuracy of a clock is the uncertainty or error in that frequency. A measurement of the difference of two clocks for accuracy, can only show a lack of accuracy, if the slope of the difference is not flat. But flatness doesn't prove accuracy.
Ultimately, since the second is defined in terms of a transition in the Cesium atom, only a Cesium clock can be accurate. We evaluate the accuracy of the primary Cs. clock for the US by first creating a table of all the potential (known!) sources of inaccuracy, then we measure the magnitude of each potential source in at best two or more ways. For a clock not based on Cs., one can do the same things, but the theoretical rate cannot compare with the theoretical Cs. rate accurately, simply because we cannot evaluate the quantum mechanical equations with enough accuracy to transfer the difference theoretically with enough decimal places.
There are now "accurate" atomic clocks based on Aluminum, Ytterbium, Strontium, and Mercury. Since these have the potential for more accuracy than Cs., eventually one of them will be chosen as the new definition of the second. Measurements against Cs. will be used to define the new rate. The primary Cs. clock is accurate to 5 x 10 to the power -16, i.e. to about 16 decimal places. The new clocks are expected to be accurate to about 18 decimals. So the first 16 decimals will set by measurement against the Cs. standard.