Deleting data would kinda go against blockchain, no? Doesn't the immutability of the data what makes blockchain secure?
At the same time, I think storing private data and removing can be possible on layer 2. It is for Apps to develop and come up with solutions. For example, Hive has 'custom_json' operation where we can store any kind of text data.
Sometime I wonder if following scenario is possible. We could build an app that would encrypt the data and store the data in the encrypted form. Only user would be able to access that data in a meaningful way. For everybody else it would be some random text with no meaning. When we need to delete the data, we destroy the link between the data and decryption. That way nobody, even the original user can access it anymore.
It kinda like how computers work. When we delete a file on our computer, data actually is still in the hard drive, by simply clicking delete we only removed the link to the location of the data on the hard-drive.