In my opinion, there is nothing wrong with checking your partner's texts.
If you have self esteem issues, then you are likely to be curious. If you are in this position (I was) and do check, then there are two possible outcomes;
either:
You find nothing and realise you were being a bit silly, probably irrational and you learn to trust your partner because you have no reason not to;
Or you do find something incriminating, in which case you were right to be suspicious and they have questions to answer.
If a relationship is really that strong, you should be able to talk to members of the opposite sex without crossing a particular line (which may have been agreed). If you feel the need to hide all your conversations, you probably shouldn't be having these conversations.
Say you go to a café with a member of the opposite sex without your partner knowing, they walk in to grab a coffee before work and see you with the other person. If you're unaware that your partner is there, does it put them in the wrong if they overhear you and this other person being inappropriate? Not really - you wouldn't be angry at them for overhearing. I know it's a different situation and not a great comparison, but I don't understand why people feel like their texts need to be so much more private. I would have no problem with a partner checking my texts because I'm confident that they'd never find me being inappropriate with other people.
I don't think someone who is being a lousy partner can be angry for getting caught out. I think the "respect my privacy" line is an attempt to play down the seriousness of what the person has done a lot of the time. If everything was so innocent, who cares if people read it?