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Yeah, I feel similarly, although I know some conversations should be "off-limits" e.g. because they deal with someone else's issues. So, for instance, my wife will tell me "please don't read X chat, because that's where Marie's venting about her issues" and I think we should respect not because our SOs' privacies, but other people's privacies.
It's an invasion of privacy ofcourse it's wrong, if you look through their private messages without their permission that's very very bad
They're called private messages for a reason >

For the beginning of any relationship, I agree. It is weird and presumptive to insert yourself into the conversations and relationship that your partner is having unless and until they invite you in.
That said, as someone who has been married to his partner for 8 years (and in this relationship for over a decade), it feels curious to me to think of a long term relationship in which there are forbidden conversations that your partner is having that you aren't allowed to hear. My wife and I pick up each other's phones all the time, mainly because we find the texts that we send and receive hilarious. There is nothing in there that we wouldn't say to each other's face, and if our friends want to say anything private to us they would do so in person or over a phone call. 
Obviously different people have different relationships that can be healthy, and maybe this is partially a generational thing as someone in their upper 30s, but it is personally hard for me to imagine a healthy decades-long relationship that I would be in where my partner wouldn't trust me to look through their phone for amusement.