"press independence" is correct and usable in written English.
You can use it when referring to freedom of the press or media freedom. For example, "The preservation of press independence is essential for a functioning democracy."
Press independence is as essential a bulwark of political liberty in Britain as it is everywhere.
The new president has promised to restore freedom of the press, independence of judges, and the rights of religious and ethnic minorities.
Congress has supported the use of confidential sources and press independence before, as in its vote on a CBS News program, "The Selling of the Pentagon," in 1971.
Instead he offered us Singapore — which ranks 152nd out of 195 nations in the Freedom House ranking of press independence — as "the most open and clear society in the world," where government ministers are paid so lavishly they have no temptation to be corrupt, and thus presumably there is no need for a nosy press.
Does he want to be remembered as the man prepared to sabotage parliamentary bills and sacrifice three centuries of press independence on the altar of political opportunism?" The Daily Express weighs in with an editorial that might have been penned by Beaverbrook 60 years ago: "There lurks in the socialist soul a sense of grievance and desire to shackle a newspaper industry…" Socialist soul, eh?
After World War II, Burundians began to press for independence.
After World War II Burundians began to press for independence, which was achieved in 1962.
This is bad news for journalism, press freedom, independence, the whole shebang.
Only through the growth of advertising did the press achieve independence".
Ethnic, linguistic and sectarian communities may continue to press for independence, and no doubt the Palestinians and Kurds deserve it.
God forbid!" Youtube 6/10 North Korea – Yes The North Korean press believes independence "would be a very positive thing for Scotland".
