As a practicing MSE specialist I'll add my two bits here. My own opinion is that both structural and geotechnical expertise is required. Here's why.
It's important to remember that the design rules for MSE walls (FHWA, AASHTO, BS8006, Clouterre) are simplified methods which have adapted the theory based on empirical observations. If you don't understand why, for example, the state of stress within the MSE mass varies (and differs depending on the type of reinforcement) then you shouldn't be designing a MSE wall. An engineer with at least a solid grounding in soil mechanics is best placed to make judgements about the behaviour of the mass when one strain rate is supressed. Also the design rules, because they are based on empircal observations fall apart when applied to novel cases. A geotech is likely to have more sophisticated tools to evaluate the overall stability of MSE structures in unusual applications (three or more tiers is a great example.)
On the other hand geotechnical engineers tend to be clueless about practical aspects of MSE wall design. Ask a geotech about the facing panel design and you are likely to get a blank stare. Structural engineers also tend to have a better feel for evaluating materials, loads and their application to structures such as parapets and anchor slabs and performance criteria when these criteria are driven by the supported structures (pavements or bridges).
I guess where I tend to end up on this issue is that a structural engineer is usually perfectly competent to handle MSE wall design or checking when he understands the basic theory of MSE walls, the MSE application is standard and incorporates standard materials, and the wall is founded in very competent soils.
In all other cases the structural engineer would be best advised to get specialist geotechnical input.