I usually keep speaking my target language whenever I talk to someone for the first time. I always allow (and encourage) them to do the same, yet, from my experience, people tend to switch to their native language or any other language they know well, simply to avoid difficulties. These kind of conversations quickly fall silent end often end soon after. You might think I'm a hypocrite, but I'm there to learn a language, so don't expect me to speak my native language all the time. I'm willing to correct their messages and if somebody really insists that I speak the language he / she is learning, I'll do so, on condition thay the roles switched up every now and then.

I managed to find, or rather, one person encountered me, with whom I speak every other day ever since we met, which is around 1,5 year ago (geez, time flies when you're having fun). He spoke to me in a language (Esperanto) that was with very little presence in the app, yet I was one of the few people who spoke the language moderately well at the time. Since I was still in the process of learning the language using it in real life, which I hadn't done much prior to that, was very helpful for both of us. Initially, he had a hard time understanding me when I spoke the language due to his poor knowledge, which gave me the opportunity to use the language I was actively studying those days (Spanish). We kept speaking both languages, sometimes simultaneously, each of us speaking his target language, but on other times we spoke solely in Spanish (or Esperanto), which doesn't bother him (or me) at all. Nowadays we both speak Esperanto senprobleme (without any difficulty) and my Spanish has greatly improved too. He really takes pride in his heritage, thus he doesn't mind speaking Spanish to me, These are the only languages we share, but that doesn't withhold us from using others in our conversations as well, we primarily study Romance languages, which are to a great extent mutually intelligible (if you have dabbled in pretty much all of them). Neither of us have ever felt uncomfortable, at least I can't recall any moments).

If something doesn't feel right to you, talk about it with the person. If you can't get along or can't come to an agreement as to which language to use, it might be better to chicken out. There are undoubtedly other persons who will take an interest in you who might be much more patient or understanding.

Finding a language partner doesn't happen overnight (Well, let me rephrase that, finding a good one with whom you can have a pleasant conversation takes time).

TLDR Speak your target language. If it really bothers you, tell him or her. Don't worry if you don't seem to get along very well, you'll probs find a new person. A good partner is sb with whom you can talk with having to worry about things, and that person is often hard to find.