I think ArcGis Pro gives you a lot more control over transparency, with proper alpha masks.
But you can still get a similar effect in normal ArcMap. It's a little bit of work, but once you've done it once you can do it pretty quickly.
1. Change the raster to "classified", with five classes
2. Fiddle with the class breaks to get the highlights and shadows looking right. You want at least two highlights (bright and really bright), two shadows (dark and really dark), and a neutral class that will represent flatter areas. You could also do this with three or four light and dark classes, but it's more work for not much more visual benefit.
3. Now change both highlight classes to white, and both shadow classes to black. Change the neutral class to no colour.
4. Duplicate this layer four times. You'll probably want to group these into a "hillshade" layer group as well at this point, for neatness' sake.
5. Name the layers "lightest", "light", "dark", "darkest".
6. For each of the four layers, null out the colours that aren't being used. So "lightest" would have a white at the top and then four nulls, and so on.
7. Now set the transparency for each layer until it looks good. I'd typically use something like: lightest: 80%, light: 90%, dark: 95%, darkest: 90%, but it depends on your underlying imagery/map.