To understand how training on conversational language impacts performance of pre-trained models on downstream dialogue tasks, we build compact Transformer-based Language Models from scratch on several large corpora of conversational data. We compare the performance and characteristics of these models against BERT and other strong baselines on dialogue probing tasks. Commercial dialogue systems typically require a small footprint and fast execution time, but recent trends are in the other direction, with an ever-increasing number of parameters, resulting in difficulties in model deployment. We focus instead on training fast, lightweight models that excel at natural language understanding (NLU) and can replace existing lower-capacity conversational AI models with similar size and speed. In the process, we develop a simple but unique curriculum-based approach that moves from general-purpose to dialogue-targeted both in terms of data and objective. Our resultant models have around 1/3 the number of parameters of BERT-base and produce better representations for a wide array of intent detection datasets using linear and Mutual-Information probing techniques. Additionally, the models can be easily fine-tuned on a single consumer GPU card and deployed in near real-time production environments.