Maybe not “mutually exclusive”, but there are conflicts. For example, some EU countries media are traditionally more reticent about publishing identities of the arrested than the American media is.

The most obvious recent conflict is “Right to be Forgotten” laws in EU vs the U.S.

Well, can you name an example where the EU focus on privacy instead of free speech had an actual negative effect?

Because as I see it, witch hunting old people who fake recipes for their opiod addiction or adding "massive national fame" to the list of motives for school shooters doesn't seem to benefit anyone beside the infotainment industry - and anyone who doesn't like actual coverage instead of sensationalism.

The very essence of free speech seems to be equally there in the EU (if not more) as it is in the US, or am I mistaken?

Anything that requires a takedown will have negative consequences depending on who you are:

https://www.theguardian.com/commentisfree/2018/mar/05/right-...