The Ginzburg's paper quoted above has a specific setup which is related to the examples like the "preprojective algebras" related to quivers. There is a recipe attaching an algebra to a noncommutative algebra by generators and relations reminiscent of the Weyl algebra. This recipe is local not only under flat localization, but even under stably flat maps. Yuri Berest has studied these aspects with hopes to globalize that definition to the nonaffine situations.

The usual Grothendieck definition may work in some noncommutative cases, e.g. possibly when the only noncommutative variables are nilpotent and alike. I do not know which definition in Kapranov, Oren means in his comment, but I think he points to similar cases related to nilpotent thickenings. Already for quantum groups this does not suffice.

Lunts and Rosenberg have tried to find a definition which would go along the Grothendieck's geometric picture: dealing with resolutions of diagonal. This has been studied in two of their Max Planck Bonn preprints, in very abstract categorical framework, and the results are global in the language of categories of quasicoherent sheaves on noncommutative schemes so to speak. Then they wrote two other papers on the same topic with down-to-earth recipes in the case of rings, modules and graded case. These differential operators correspond to filtrations reminding the Grothendieck's case of filtration by order, but being corrected in an improtant subtle detail. Their basic property is standard behaviour under flat localization functors as you suggest. There is also an arxiv paper by Tomasz Maszczyk who uses a variant of nc algebraic geometry based on bimodules and monoidal categories, and he rederives the same definition of the ring of regular differential operators as Lunts and Rosenberg do, with different geometric insight based on the duality between the infinitesimals and differential operators.

For the references look at nlab page on differential operators in nc geometry which I just started writing

nlab:regular differential operator in noncommutative geometry