Here is the secret: Don't use the (global) radio in the editor in MP, since it can be used by anyone (unless you sync it with a group, but then anyone in that group can use it) and when used causes the trigger to fire on everyone's machine without any way to know who actually pressed the button.
Instead, use a script-created trigger, which then is triggered locally, so it is obvious who pressed the radio button. You could either only create the trigger for leaders (bad if the leader gets killed and no-one else can use it) or create it for everyone and ignore it if the local player isn't currently leader of his group. Remember that if the radio needs to have global effects, you'll have to manually communicate these effects in this case.
Radio triggers, made in the mission editor, work great in SP. This workaround only applies to MP and only when you need to know who actually pressed the radio button.