Yes you certainly can!
If you navigate to your user profile in Open LMS and go to Preferences>Message preferences, a panel on the right of the screen will open up. On it you will see 'Notification preferences' with 'Activity stream' and 'Email'. Just ensure 'Email' is switched on and you'll start receiving these notifications via email. If you want to receive them inside of Open LMS as well (via the bell icon) then you can also switch on 'Activity stream'.
If you don't see 'Email' and 'Activity stream' in the panel, it means we need to enable that particular setting. In this case you need to navigate to Site administration>Messaging>Notification settings. From left to right you'll see several columns, each corresponding to a notification method. On the left-hand side running toward the bottom of the page, you'll find many rows corresponding to particular notification types. Find the 'Email' column then scroll down to the 'Personal messages between users' row. In the corresponding cell you can set the dropdown to 'Permitted' and tick the 'online' and 'offline' boxes then save changes. This will unlock the setting in your profile described above.