The easiest solution is using JJ2+, which supports online coop via the /coop chat command. If you're running the GOG.com release, you should obtain JJ2+ from there instead, either from extra downloads on the site or beta channels if using GOG Galaxy. After installing the mod, start the online or LAN game in any mode and level, press T to open chat, send /coop to set mode to cooperative, and then /c easter1 to switch level to the first level of the official campaign. Methods outside of JJ2+ may exist but at this point it's unlikely that anyone would want to go through the trouble of figuring them out.