What is most likely is that he spent his entire career in Mantua and was probably connected with or enjoyed the favor of the court of the Duke of Mantua. Rossi's first publication, the Canzonettas, was dedicated to Duke Vincenzo Gonzaga and its first piece is dedicated to the Duke and his Duchess. Rossi also dedicated his next publication, a book of five-part madrigals, to Vincenzo. In addition, Vincenzo exempted Rossi from the requirement to wear the yellow badge that otherwise was mandatory for Mantuan Jews. This exemption was pronounced in 1606 and Vincenzo's successor, Duke Francesco II, renewed it only six days after inheriting the dukedom. However, while he often appeared at the court as a salaried occasional musician, it does not appear that he was ever appointed to a regular position there. For one thing, the regular musicians were also members of the cappella of the Church of Santa Barbara, which both sides would have felt incompatible with his Jewish faith.
Instead, he seems to have made his career as a member of various theatrical troupes in the city. These were primarily Jewish in their personnel, but played in court and among the Christian community. He was in demand: In 1612, the Prince of Mirandola asked to engage Rossi and his musicians to come to Mirandola to play for the visiting Duke of Modena. Rossi's last published work appeared in 1628 and there is no further documentation of his history, but in 1630 troops of the Holy Roman Empire conquered Mantua, following which the Mantua ghetto was destroyed and a severe plague struck the city. It is considered likely that Rossi died in one of these sad events.
Rossi was a leader in the new Baroque styles of music and the first to publish madrigals with basso continuo parts, although otherwise the style of the five-part counterpoint found in these books is rather conservative and, in fact, he published a set of four-part madrigals rather late. He wrote light vocal music, several important sinfonias and other instrumental music, and Jewish liturgical music, such as his Hashirim asher lish'lomo (Songs of Solomon) with 33 polyphonic settings of Hebrew texts. He is credited with leading the transition from the Renaissance form of the canzonetta (with a generally homogeneous texture) into the Baroque form of the trio sonata, with its textual distinction between the upper voices and the bass line.