Do I need to stay online when the recording conversion is in progress if i have local recording ?

No, you do not have to stay online to convert a local recording, but your computer does need to remain on and logged in. Once the meeting is over and the recording begins converted, the system has all the recording data it needs, now it just needs time to convert that raw video data into a video file you can use. 

Hope that helps and please make sure to mark the solution as accepted if this information is what you needed.

What if the computer battery dies during the conversion? How can I recover the file?