Yes, as each country has different laws, the IP companies are allowed to monitor and block you, in some cases report you.
My brother in law downloaded a film and he got a letter from the IP service saying he had to stop or else.
If you are expected of committing a crime I think the law can look through your emails and text messages if they believe it can help with their case once they have arrested you that is, not by coming up to you and say I need to search your phone etc.
In the UK Gary Glitter was arrested for having child porn on his computer, how they found out was, he had taken it to PC World to be fixed and they saw the content and told the police.
It's an old story but here it is
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