For Hawkwood specifically, he was probably literate (although this has been debated) but he was not in general a man of letters or known as someone with rhetorical skill. He had agents, clerks, and scribes to handle most of his personal business and correspondence. The most important person to handle Hawkwood's letters was his second wife, Donnina Visconti, the illegitimate daughter of the Duke of Milan who he married in 1377. As an educated woman, she was literate and helped Hawkood carry out his business and interact with various Italian communities in colloquial Italian as well as Latin. Donnina wasn't alone in being a significant part of the mercenary business. Foreign mercenaries in Italy often married Italian women, whose fluency in local language/dialects was critical to enabling their husband's business activities. If the free companies were like mafias, then the Italian women who married mercenaries were the consiglieri of the crew. So that's one way of solving the language issue for the foreign soldiers: marry/hire a local who could speak or write for you.

Hawkwood himself became fluent in Italian (unsurprising given the decades he spent in Italy). William Caferro's biography of Hawkwood (where I am drawing the information in this answer from) portrays this as a gradual process: in the first direct quotation attributed to Hawkwood, he says something in French. Between Norman French still being spoken in England and Hawkwood's prior experiences fighting in France, it is probable that he picked up at least some French there. Caferro also notes that some surviving letters written by English mercenaries in Italy from this period are in a version of Italian with a strong French influence. It seems from this that English soldiers in Italy used French to bridge the gap between English and Italian, until they spend enough time in Italy to pick up more of the local language/dialect. That in turns suggests that it was not unusual for English soldiers in the Hundred Years War to learn a bit of French, although there is no evidence that would allow us to estimate just how common that might have been.

Language gaps were a common issue across the mercenary world of medieval Italy, and people fluent in multiple languages could make themselves very useful. One Italian bragged about knowing English and how that allowed him to negotiate on better terms with Hawkwood. German and Hungarian professional soldiers working in Italy from more educated noble backgrounds would converse in Latin and give commands in it, in addition to it being the standard diplomatic language of medieval Italy. Many, if not most, mercenaries banded together with others of their nationality, so that would have helped the language issue within their immediate circle.