The SUA has its own characteristics and limitations. Is there any unit responsible for providing separation, control or information, for pilots flying into or across any SUA?
In many cases, there are controllers dedicated to providing ATC services to the aircraft that are USING the SUA. Callsigns such as "Nellis Range" or "Eglin Mission" are used for the controllers working with aircraft within the SUA, be it MOA or Restricted or Warning areas or whatever. What looks like one big chunk of airspace on the chart may likely be divided into separate areas for training or testing purposes that the aircraft flying into it will know about, and it's useful (sometimes, vital) to have a controller monitoring the various missions and keeping the various aircraft apart.
These controllers may work in a dedicated facility that only deals with military aircraft, or they may work within a larger facility that includes "regular" ATC functions as well. For instance, the controllers who work the off-shore warning areas south of Eglin AFB as well as the on-shore restricted areas east and west of Eglin are physically located in the same RAPCON that is Eglin approach. So civilian aircraft inbound to Crestview or Destin or Eglin/Ft Walton Beach are talking to controllers working one set of scopes, but across the room are the controllers for Eglin Mission who are watching the various SUA. An aircraft that departs Eglin (or Hurlburt, or Duke Field) would probably be talking to one of the former initially, then when they head into one of the restricted or warning areas, they'd be handed off from "Eglin Approach" to "Eglin Mission." Essentially it's like being handed off from one Approach sector to another, just that the Mission controllers have slightly different roles in keeping the airspace safe.
It's far better to clear an aircraft transiting a particular sector of SUA to fly through it ABOVE the aircraft in it that is employing air-to-ground ordinance, for instance, than clearing him through the airspace BELOW the bomb-dropper! Or, if necessary -- maybe it's a helo with an emergency that needs a straight line return to base and isn't going to climb up very high -- at least you have the bomb-dropper go cold and hold up high until the transiting aircraft is out of the way! That's the sort of ATC that Eglin Mission does that Eglin Approach doesn't... along with confirming that this callsign is in fact scheduled for this range at this time, and the last aircraft has departed, so "you're cleared in to Range XYZ to work surface to 15,000', advise 5 minutes prior to departing."
It varies for the different types of SUA's.
They include: restricted airspace, prohibited airspace, military operations areas (MOA), warning areas, alert areas, temporary flight restriction (TFR), national security areas, and controlled firing areas.
For the U.S., if the SUA extends above FL180, and the pilot will be permitted to cross said SUA, then IFR rules exist for Class-A airspace, i.e. regular ATC will be in contact.
So again it depends on the airspace class, and type. For example flying through an SUA in Class-G airspace during permitted hours, Class-G rules apply.
Detailed information for a specific SUA will be on the aeronautical chart.