They have a few places in the kosel tunnels, including the spot closest to the kodesh hakedoshim. Near the kotel they have a place where there are actual stones f the beis hamikdash that were thrown over the outer wall-you can daven there. Also, there is a mini kosel in the old city.
I personally would only do those things on the kosel plaza. That is okay. But past the fence my phone is on silent and I rarely see people doing these things.