This will depend on your browser and desktop security configuration. Assuming each user has a separate account on the desktop machine, I would expect that each has to accept (security prompts) and install it. The PegaCALL applet is no different from any other applet in this respect.
Note that Pega leverages http-based event delivery in PegaCALL 220.127.116.11 and higher, with no need for browser plugins (applet or activex) and related challenges with desktop and security configuration and administration.
Our Customer is worrrying about the Applet download prompt each time when CCP's use the Portal. Specifically this will be prompted everytime to a set of CCP's who are accessing the Portal Remotely(through CITRIX environment).