The Twitter thing is just a new way to bypass journal restrictions. It has been going on like this for years -
1. It is perfectly accepted by journals that scientists share their own paper, under 'fair-use' regulations. Simply dropping a line to original authors will usually get you their paper without any cloak and dagger tactics.
2. Sites like Libgen have an expanding archive of papers. Running the site is illegal, but downloading material from Libgen isn't (at least in Europe).
3. Posting on the r/scholar sub-reddit will also usually get you the paper within a few hours.
4. A friend/colleague/collaborator at an institution with access to the journal will usually gladly forward you the paper.
5. Many smaller institutions maintain collaboratory VPN access to larger institutions. This allows you to piggyback on their subscription.
If these scientists using regular unencrypted email, it is hardly private. However, the reason the journals will never come after individual scientist is that scientists are their content generators. Any journal suing a scientist will immediately be boycotted and will go out of business. Instead, journals try to make it harder to download papers, while letting scientists read them if they have a subscription (eg: Readcube). Of course, this is a fool's errand.