Doppler is a very, very sensitive beast.
There are three main ways of getting errors:
1) You must be in line with the direction the target is moving in order to get an accurate velocity measurement. The element of the equation is COSθ (θ = angle between your line of sight and the actual direction of travel). COS90 is zero, so the more out of line you are, the greater the error. This error always reduces the velocity and is in your favour.
2) Overshoot - scraggly crap around the top of the Doppler trace. It takes experience of looking at a Doppler signal to differentiate between overshoot and an actual peak velocity. I would not trust a machine to do this accurately. Sometimes to make the trace clear, you have to play with gains, post processing and the colour scheme of the graph to make it more readable. In medicine we never trust a machine to automatically trace a Doppler signal.
3) Angle of the surface - as you say, it needs a flat surface that is not angled or anything to allow the beam to bounce back to its point of origin. If it doesn't, then you'll get scatter and the trace will be bullshit. This is how radar invisible bombers work, by not having right angles and so, in conjunction with radar absorbent materials, the beam is absorbed and scattered rather than returned to the source.
I have spent ages staring at and playing with Doppler traces when I've had half an hour acquiring the perfect signal. I would not trust someone to acquire, using a hand held piece of kit, an accurate peak velocity, with a clean signal that can be easily measured by a computer on a target moving at 8m/s (30mph), never mind 68mph. Given the range (bear in mind a 1 degree wobble at the source is going to be a massive wobble at a distance), the size of the target (a numberplate) and the speed, not even a qualified marksman or army sniper could be expected to make that shot reliably.
I'm sorry but just basic physics and experience of shooting things tells me that the manufacturers of these things are playing on people's ignorance and not being honest about their limitations. I'm sure the police use them in good faith however, they're not going be trained to understand the physics involved and therefore the limitations of the readings they acquire. I feel that the machines should record the exact trace and that this should be submitted, alongside the video, for analysis by a trained person before a fixed penalty can be delivered. It would take maybe 5 mins per ticket to say if something is reliable or not. The current system relies on a computer to make sense of data collected in imperfect circumstances and is used to criminalise people on the spot.
EDITED to stick the theta symbol in properly.