Trailing Stop works in the client terminal, not in the server (like Stop Loss or Take Profit). This is why it will not work, unlike [Stop Loss or Take Profit], if the terminal is off. In this case, only the Stop Loss level will trigger that has been set by trailing stop.
source: Trailing Stop - Trading - MetaTrader 4 Help
Therefore, if you want to have your Trailing Stop work even if your platform is not logged into the account in question, you must either run MT4 on a virtual private server (VPS) or use a different trading platform that offers server side trailing stop functionality. As we pointed out the other day, one platform can’t please everyone.