When you send or receive calls from your phone, MightyText can store the call logs in your account.
As of February 2013 (in MightyText Android App v 3.35)
By default, MightyText will notify you on the web app, or the tablet app, but will not store call logs.
If you would also like to have the call logs stored on MightyText, then please follow these steps:
1) On your Android phone, launch the MightyText App (v 3.35 or later), and launch Settings
2) Make sure the checkbox for "Phone Calls" is checked
3) Tap "Store Phone Call Logs" and choose "Store Call Logs"