A Twitter Mobile App featuring SSI would require all users of this social network (Alice, Bob and even Mallory) to create self-sovereign identities, register them in one of the publicly-available SSI registries (DID registry) and then link their SSIs with Twitter accounts.
In this case, Alice would have her private key stored securely on her device, so she could use it every time she wanted to sign her Tweets before publishing them on Twitter. Bob’s Twitter App with built-in SSI support would then verify the signature and confirm that the Tweet really comes from Alice. If the hacker Mallory wanted to scam Bob by sending a Tweet from Alice’s account using Tweeter’s internal tool, she wouldn’t be successful! She could still send him the Tweet, but Bob’s SSI App would spot right-away that the Tweet is either unsigned or worse, not signed by Alice. Thus Bob would have never given away that 1 BTC to Mallory.