What does "private empires" mean?

not literal empires, but separate power structures within the larger government system.  They are "private" in the sense that there was a particular individual in charge of that group, who used that group to do things and pursue goals that were defined by the group leader.  Himmler was probably the biggest operator of a mini-fiefdom in that government.  He had his own private army, in effect, but he did have power and control rivalries, especially with Goering.  Palace intrigues.  Which individual had the great Adolph's support today?

It means that within the nazi regime there were those who were running their own 'sub-regimes' or their own private regimes to forward their own agendas, but knowing and understanding that Hitler was the total authority over them, and could stop them any time he wanted.

Government is supposed to be a public affair, thus owned by the people.

Devolution towards privatisation means that public property, such as car parks, or trains, becomes the property of private owners, that no longer require to redistribute the profits made from trains amongst the people, and can give tickets or charge for parking, for personal profit et cetera.

Therefore a private empire would be when privatisation has gone so far, that everything that was once public owned, is now in the hands of a few corporate tycoons, or even one single private owner, that could even be considered a monarch.

The empire is no longer owned by the government and people of that country or countries, but instead by corporate persons that might not even live in the same country they are sending parking charges too.

Also with power in the hands of corporations, or companies, that is also anti-democracy, as company owners and monarchs do not take votes from the people, and their decisions are more along the lines of dictators, and company policy decides what decision they will make, not the vote of the people.