The ん in your two examples are different, the first being part 何 while the second is a contraction of の, but the general meaning of だろう・でしょう is the same in both. If だろう or でしょう is paired with a question word or followed immediately by か, it's basically a way of asking a somewhat self-directed question, like 'I wonder ...' (Note that they can also mean that without either of those things sometimes.)
In your first example A says, 'Hey, you should write a little more neatly. What does this say?' B replies, 'I wonder ... I can't read it either.' (Okay, I meant to write this earlier, but maybe B wasn't the one she was directly talking to? Seems a little weird that B wouldn't know what B themselves wrote, but then again my own handwriting is messy enough that I have trouble reading it sometimes. Dyspraxia is annoying)
In your second Ena says 'I wonder what she's doing ...' (Sequences like るんだ, るな sometimes contract to んだ, んな)
I am very tired writing this, so please forgive any messiness or typos.