So, _hypothetically_, what if you don't use a third party email provider? That is, all your mail is delievred directly to you and is stored on your local machi

So, _hypothetically_, what if you don't use a third party email provider? That is, all your mail is delievred directly to you and is stored on your local machines/devices. Does that give you more privacy protection than keeping messages stored with some third party in the cloud? Or does it give you less protection?