During a conversation, neural processing of speech production and speech perception in both speakers temporally overlap and is continuously modulated in real time, based on conversation dynamics, context, expectations, and quality of the interaction. Recently, a growing interest in the neural dynamics underlying interactive tasks, in particular in the language domain, has mainly tackled the temporal aspects of turn-taking in dialogues. Beside temporal coordination, an under-investigated phenomenon is the neural correlate of implicit convergence towards a shared phonetic space. Here, we used dual electroencephalography (dual-EEG) to record brain signals from subjects involved in a relatively constrained interactive task where they were asked to take turns in chaining words according to a phonetic rhyming rule. We quantified participants' initial phonetic fingerprints and tracked their convergence during the interaction via a robust and automatic speaker verification technique. Results show that phonetic convergence is characterized by a left anterior alpha/low-beta desynchronization in the speech motor preparation phase and by high beta suppression before and during listening to speech in right central and left frontal sectors, respectively. With this work, we provide evidence that during turn-taking verbal interaction, alpha and beta oscillatory dynamics are linked to the coordination of the " how " rather than the " when " interaction takes place. We suggest therefore that speech-based interaction, besides affecting the temporal pattern of turn-taking, leads to a critical change in speech phonetic targets. These changes in how we generate the speech output, are reflected in specific preparatory neural markers demonstrating that during a conversation, speech perception and production are highly interdependent and co-constructed by participants.