The training I assume you're asking for requires statistical training far above the intro to stats level. So if you want to do good testing that psychologists do that we MDs don't such as an MMPI, no. Unless you've taken the appropriate statistics courses and then you have to get the training for the specific tests.
Many places dumb things down and say only psychologists. Can an MD with the right statistics and other training do it? Yes but because the psychology training has it there anyways they just often times on their list of required criteria to administer the test they say psychologists only. The term "dumb down" is in no way shape or form meant to throw shade at our psychologist-colleagues. High level statistics is tough and is leaps and bounds way more sophisticated vs an intro class. Further I've seen a lot of psychiatrists think they know what they're doing and make up a lot of BS claiming they know more than they really know. On the other hand I've seen some MDs who have mastered the highest orders of statistics, know psychometric on the order with the best psychologists but even they will have the psychologist due the test because they want to follow the rules on paper.
This can become a legal pickle. Even if you have the right training to understand the tests many psychological tests say psychologists only. Some however are more open-minded and give a list of criteria such as "a prior course in psychometric testing." If you don't meet the requirements to administer the test, do the test, and this goes to court this can be flung into your face.
All true, but maybe a bit paranoid or overly restrictive, frankly.
I think the big issues with this are:
1. The vast majority of psychiatrists don't want to do this and certainly won't actually end up doing this. And certainly not full-scale IQ assessments like the WAIS. Thus, it will likely be completely wasted training the vast majority of the time.
2. I think we really, really need to push past the 1940s,50s,60s,70s, 80s MMPI mystic/allure and move-on with more tailored, robust, simplistic, and/or outcome-driven measures for diagnosis and treatment planning. "Empirically keyed" measures are great, but only have so much ROI. This may just mean moving back to more skilled structured interviewing/investigation and assessment, as well as looking at other allied health assessments and markers of psychopathology. There will always be a place for the MMPI and alike (especially in forensic cases/questions)....I just don't think doing *more* of them is the answer (we already do tons of MMPIs that probably add very little to treatment in the end), or will be at all helpful for the average clinic patient.
3. Everyone needs to be careful about "muddying the waters" with too much speculation, too much information, and/or (sometimes) too much psychobabble during your assessment of the patient that may not actually/functionally change the treatment plan for that patient. Right? The overuse of psychological or personality testing can easily do this.