How to approximate the sample rate?
In my experience, sound card sampling clocks vary by as much as 10 Hz from one card to another, even between computers of the same make and model. I have never seen much jitter, but I haven't especifically looked for it.

Note, however, that it is essentially impossible to know the true sampling rate, unless you have access to an atomic clock. What you can do is approximate the difference relative to a clock that you designate as your reference.

A simple experiment is to generate a single-tone signal, then receive it and find its frequency. Say your reference is the transmitter and it's operating at 48,000 samples per second. Feed the transmitter a sequence 0, 1, 0, -1, 0, .... This will produce a sine wave of nominal frequency 12,000 Hz. In the receiver, you sample this signal for a relatively long time (to have good frequency resolution) and use the DFT to determine its frequency to be, say, $f_2$. Then, you can calculate $$\Delta=\frac{12000}{f_2}$$ which is the factor by which the receiver's clock is faster than the transmitter's. Of course, this is only one of many possible approaches.