When training climbing roses onto a wall without the use of an arbor, how much space do you leave in between from wall to wires?
You only need enough space between the wires and the wall so you can thread the ties behind the wires.
It might be easier to use a wooden frame like these pictures, instead of wires. You only need to fix the frame to the wall at a few places, instead of fixing both ends of every wire.
Usually, largeish eyelet or vine screws(https://www.amazon.co.uk/VINE-SCREW-ZINC-PLATED-75MM/dp/B0041OABD2 for instance) are used for this purpose; for a brick wall, it's usual to drill into the surface and use a rawl plug (not sure what they're called in USA) and screw the eyelets into those. Because the eyelet part of the screws naturally stick out from the wall surface, there's a gap between wall and wire of half an inch to an inch, depending on the size eyelet you use. For a Rose, half inch is fine, assuming you're not intending to thread the canes behind the wire, but, ensure the wire is strong, or heavy guage, before letting a rose grow up, or the weight of the rose canes over time will break it.