I assume this question without any keylogger help installed on the local computer. So is it possible that a remote user catch the password you have typed in on your local computer for a local soft ?
No, it's not possible for a remote user to sniff your password over a service like logmein. There are many reasons, the biggest of which being that you aren't connecting to each other directly. When you connect with someone over logmein you are going through a server which brokers the connections. If a remote user shares their desktop you could not record keystrokes, only see what is on the screen.
The only way you would ever be able to get their keystrokes is if they were acting on your system, as in you had given them control, and even then you'd need a keylogger utility of some sort.