I am reading Bryan Peterson’s book, Understanding Flash Photography...

Just my humble opinion, but not the source I'd go to. Peterson is more of a landscape/nature photographer and his knowledge of flash is pretty antiquated (hence your reliance on guide numbers and distance scales). I'd recommend starting with something like Neil van Niekerk's website or books instead, or David Hobby's Strobist website and Lynda.com tutorials. These guys kind of rely on flash for the work they do (van Niekerk's a wedding and portrait photographer; Hobby was a photojournalist and does editorial work).

... I own a Yongnuo 600-Ex-RT speedlight and YN-E3-RT transmitter. When the speedlight is attached to the camera, in manual mode, I can see the distance scale on the flash. My understanding is that, if my subject is within that range, for the given aperture, ISO, shutter speed, I will record a correct flash exposure.

Well, kinda. But actually what it means is that if you're shooting direct flash from on-camera at the subject, they should be at that distance. Not within that distance. Nearer than that, and they might be overexposed in flash. Farther away, and the flash may be too dim.

And that distance won't be accurate if you're bouncing the flash, or using a large modifier, because both of those techniques eat some of the flash power. This is why autothyristors and then TTL were invented. Guide number calculations and distance is a relatively simplistic model that doesn't account for other scenarios.

This is why you should learn to use eTTL. On-camera shooting with eTTL tends to be good for events and other situations where you may only have one chance to get the shot. eTTL does the distance calculations, using the focus distance in the camera as well as iso and aperture settings from the camera, and adjusts the flash's power for you automatically. And it's going to do it a lot faster. Tools like guide numbers and the distance scale were more used back in film days when there was no eTTL and not all flashes came with autothyristors. But mostly they were used as guesstimates.

Now when I connect the transmitter, I lose this distance scale. The transmitter’s display doesn’t show the distance scale. All it allows me to do is to control the power of the speedlight.

Yes. Because that distance is about the distance of the flash from the subject. Once the flash is off-camera, it's harder for you to judge, since it's not simply "how far away am I from what I'm shooting?" but "how far away is my light stand from what I'm shooting?" Add multiple lights at various distances into the mix, and lighting ratios, and you'll start going off your rocker trying to control your flash power by distance/guide number.

In this case, how do I know where to keep the speedlight to record a correct exposure?

Shoot, chimp, and adjust. Repeat until you get where you want to be. Practice will reduce the number of times it'll take you to get where you're going. It doesn't take long to "see" what 1/4 power vs. 1/2 power will give you any more than it does to visualize the differences between f/1.8 and f/8 or 200mm and 20mm.

Or, if you have to get it right the first time, every time, you could learn to use TTL like Joe McNally, and know how to drag your aperture and iso as well as your shutter. :)