You might tell him and ask him not to read? I don't know if that'll work for you two.
For me, it's been an adjustment since me and Mya started a relationship to have her read everything I write. Alec knows that I write but he isn't interested in reading and I don't really want him to, either. But Mya and I met through the place where I do most of my journalling, and we started our journal here together. We have talked a lot about the good and bad that brings. On one hand, I have no secrets, so it "doesn't matter" if she reads the stuff. On the other, it is different. I don't censor myself, because writing is really important to me, and because we have agreed we don't want our relationship to have an impact on what we can write. But there are issues such as timing: if it's heavy news and she's already tired I would wait a while to tell her but since I write it, she will know when she happens to read it. And often it would be important to communicate about it quite soon but it's not always possible since both of us have a life and we're in an LDR. And sometimes there can be misunderstandings, which may not have happened when talking, and which can cause hurt feelings until they're cleared up (which can be a while because of the timing thing). Then again, this has also been useful. We communicate about everything. That's both good and bad, but so far mostly good. It can cause overanalysing, and unnecessary worrying. But it also makes our partnership really close and honest. I think it benefits us because our communication styles are quite similar, and therefore misunderstandings are quite rare; and also because we enjoy talking every little thing to death.
But I still would not want the same with Alec, because I think the negative effects would outweigh the positives.