No, the goal of these security products is to defend against the government, not a random guy. In that context, it's extremely important that their server undergo the same level of cryptanalysis.
We already know the system is hopelessly vulnerable to server side MITM attacks, it makes no effort to defend against that attack model. It's mentioned in the comments that they might do manual key verification in the future, but that doesn't happen now. Compromise is silent.
That only protects against a MITM between peers who have communicated with a "secret chat" previously, not two fresh peers. As "secret charts" are disabled by default it's not really a defence against infiltration; users will presumably only enabled the "secret chat" mode when they have something sensitive to talk about.
When they do enabled it for the first time, we can instantly MITM them using the attack against the "image verification" I mentioned lower down (https://news.ycombinator.com/item?id=6932053), and we can assume that the conversation is worth our while listening in on. The user will hopefully expose themselves in the belief that they are safe, and the game is over.
It's simple unauthenticated Diffie-Hellman key agreement, which is known for MITM attack. Yes, you ask A to accept B's identity upon key exchange, but to what extend A would know B is really B not the server playing along? A plausible method would have A and B exchange certificates separate from the Diffie-Hellman key exchange process, and use those as the identity verification mechanism.
Not only is it possible, they are doing it already. I installed telegram on two devices (android and ipad) and they somehow were both able to decrypt incoming messages. How did the second device get the key..?
Ah! You were mistaken in the functioning of the service (I thought this might happen). You have to specifically ask for a secure chat with a button press, normally everything is effectively plaintext.