First to be clear: Designer mode is an integrated development environment. Whereas, among other tools, the Debugger is add-on tool. This tool allows debugging another running application. Thus to debug you need a Debugger and another running application.
The Debugger can debug other applications through the network via TCP/IP.
So there are two parts:
1. To run a Debug mode you need a Thick client to be installed.
2. To run a Thin or Web clients the Thick client is not required.
Note that the Web client supports debug mode starting from the 1C:Enterprise platform version 8.2.11