streets with different name on each side

You can tag the streetnames as name:left and name:right, to show they only apply to one side of the road (where left and right is relative to the direction of the way). These tags will probably not be used by (most) renderers, but are used by some search tools, eg Nominatim.

It might be helpful to also add a name tag, with whatever the "main" name is (which will be rendered). Though if both names are of equal status, its debatable what this should be - you could list both names in this tag.