Trust me, I struggle with the same insecurities in my relationship. My girlfriend is the type of person that NEEDS friends outside of me. Before I hated it, sometime I still don't like it but I've grown to trust her. Going back a little bit, we had a fight about it (her with different guy friends and me with 1 friend that was a girl). We had a resolution that I hate but so far has worked pretty good. We both have access to each others phone, no questions asked.
I don't mind it because I have nothing to hide, and she feels the same, we both think it's childish though that we both have these insecurities that lead us to this. We have also decided that there is no "silent time". If there is an issue, say it and we will try to resolve it. She suffers from severe depression, I'm not as bad but just like literally everyone in our age range, I have some sort of depression.
These two things have helped us out A LOT. Honestly we don't go through each others phones much because we've grown to have trust in each other.
At the beginning though, things got a little tense. She was also sending / receiving flirty texts. I did some stupid things to find out what I'm about to tell you, but long story short, she isn't one for confrontation. If someone sends something flirty she won't shut them down because she feels bad which kinda makes her send one back. After the person who told me that had told me that, I started looking at the conversations I was worried about, and they were right. The guy would say something flirty and she wouldn't 100% cave in like let's hook up, but more just to keep the conversation going so she could talk to someone.
Another thing we did was explain the situation to the person either I or my girlfriend was worried about and we would go on like a 3 person date. It made us feel better because we knew that the person knew we were in a relationship (granted it doesn't stop people), and has actually made a couple friendships.
My advice to you is first to talk to her. Explain that you're worried about your position in the relationship with this other guy talking to her. I would ask if you could see the texts before saying you already did, but then explain what part of the texts bother you. Feel out where that goes, maybe if need be see if she would be willing to take the other guy out with you.
I know how I'd feel if I asked my girlfriend if she'd go out with some guy if she were single and she said yes, but the fact is, it's all hypothetical. She's single. So either you haven't met yet or broke up so it's ok for her to say he's a good enough guy that she would consider going out with IF she were single.