A whisper is only visible to you and the person it was sent to.
Private message This option will send "private chat request" to the person selected and will show in their "alerts".
View Profile This option only show for registered members and will take you to their profile page.
Ignore Ignore option allows you to not see any messages sent by the person you have ignored.
Other Rest of the menu show some basic information about that member.
To get to this page click on top right hamburger button, then select settings.
What happened to old chat room and why introduce a new chat software?