Can I create a good Speech Recognition Engine while having millions of recorded conversations?
Yes, having lots of recorded conversations is great for building a speech recognition system. You will still have to create training samples (Each sample will be parts of Wave file --> text), but you will need lesser number of samples. 

High level steps are :


Train a GAN on raw audio 
Train a Language model on raw text data (it needs not be from these conversations, but has to be from same domain). For example, if conversations are related to medical, train language model on medical text. 
Merge these models and train on labeled samples 


For step 1, Google WaveNet is a good example (it is eventually used for Text-to-Speech, it is a component in Speech-to-Text as well)

https://deepmind.com/blog/wavenet-generative-model-raw-audio/

Papers that cover design and overall approach : 

https://arxiv.org/abs/1711.01567
https://arxiv.org/abs/1803.10132