I did it last year during post-doc process. From how they did it then, they attempted to match you based upon specialty and geography. I ended up getting a call from someone interested after I secured a post-doc, and they seemed quite interested, even though I was only half-way done with internship at that point.
The VA puts a LOT of time and money into training with the expectation that they will get out of it what they put into it. However, as mentioned before, many trainees leave the system. The general vibe that I got is that the VA is very interested in retaining people (especially when they have the basics down and could be up and running quickly).
The downside is that the places where I got matched to where geographically less desirable and not my target locations (e.g., think rural CBOCs and the like doing generalist work), especially without the specialist types of training that may be more desirable for larger/more popular VAs.
Part of me thinks that it can always be helpful to have more interviewing experience, and to try it out to see what it is like. However, make sure that if you do choose to go into it that you go into it 'authentically' or being able to say yes to the right opportunity. Also, other thought with this is that it may take longer for them to get you onboard as a staff psychologist than as a post-doc, which may leave time in for a break of sorts before you are hired, which could be good or bad depending on your situation. Just another thing to think about it. Others here may be able to speak to that themselves. I know I have heard of a wait time (as much as 6 months) while others moved in right after internship/post-doc without an issue.