Conversational bilingual speech encompasses three types of utterances: two purely monolingual types and one intra-sententially code-switched type. In this work, we propose a general framework to jointly model the likelihoods of the monolingual and code-switch sub-tasks that comprise bilingual speech recognition. By defining the monolingual sub-tasks with label-to-frame synchronization, our joint modeling framework can be conditionally factorized such that the final bilingual output, which may or may not be code-switched, is obtained given only monolingual information. We show that this conditionally factorized joint framework can be modeled by an end-to-end differentiable neural network. We demonstrate the efficacy of our proposed model on bilingual Mandarin-English speech recognition across both monolingual and code-switched corpora.