Steve Hughes describes how Power BI data classification works:
Power BI Privacy Levels “specify an isolation level that defines the degree that one data source will be isolated from other data sources”. After working through some testing scenarios and trying to discover the real impact to data security, I was unable to effectively show how this might have any bearing on data security in Power BI. During one test was I shown a warning about using data from a website with data I had marked Organizational and Private. In all cases, I was able to merge the data in the query and in the relationships with no warning or filtering. All of the documentation makes the same statement and most bloggers are restating what is found in the Power BI documentation as were not helpful. My takeaway after reviewing this for a significant amount of time is to not consider these settings when evaluating data security in Power BI. I welcome comments or additional references which actually demonstrate how this isolation actually works in practice. In most cases, we are using organizational data within our Power BI solutions and will not be impacted by this setting and my find improved performance when disabling it.
As Steve notes, this is not really a security feature. Instead, it’s intended to be more a warning to users about which data is confidential and which is publicly-sharable .