Can a member of the ELL community propose/ ask another member to share their email address to further communicate the ideas about the English language with one another privately, not in the chat room, devoid of privacy? Since nobody is being forced and there's a will, isn't there a way?
I think the easiest solution for you is to make a junk or temporary email address, give that email to the person you want to talk with and then switch to your actual email address when they send you theirs.
There's nothing wrong with proposing discussing things away from the site but I don't know that users are necessarily going to be interested in chatting over email. You shouldn't pressure them to do it and they may consider you rude for asking. Many people are disinterested in giving out their email address, so I recommend you follow this path as it gives them the chance to decide if they want to contact you first rather than you asking them for their email.
Another option is to make a gallery chat room and only give this other person permission to chat there. It will be visible to everyone but it will only be you and they (and moderators) who can chat there. If you just want to talk about a question or English in general, it may be easier to do this.
There is no way to privately convey information on the site.