It is interesting that you would mention Florence because they produced one of the more notable coins of the time, the gold (and later silver) florin. The florin became an international currency at the time (1252). It was valued really around the world for its pure gold composition and artistic designs. It appeared at the right time as well because previously the Byzantine solidus was the currency of intentional trade. The Byzantine Empire was stalled at the time between the Fourth Crusade and the Reconquest of Constantinople. Other Italian City-States, such as Genoa, Pisa and Siena tried to mint the same quality coin however it was clear the florin was the preferred currency. In 1284, Venice produced another similar gold coin, the ducat. In the years to follow many of the Italian City-State would produce their own version of florins and ducats but the ones issued from Florence and Venice remained of the highest standard and international renown.

With Italy at the forefront of the commercial revolution it makes sense that their currency would become used around the (Mediterranean) world. The coinage produced is the ultimate sign of wealth of a nation or in this case city state. Florence and Venice, as two of the mot prosperous cities in Italy were able to produce the largest gold coins. The purer and more artistic the more other places were willing to accept the coin as payment.

As for at time period of when Florence minted coins would be accepted by places under it's sphere of influence, the answer would be right away. Florence florins were one of the most secure currencies in Europe at the time and it was trading very well internationally. The only reason I would think a coin would not be accepted is unless the specific city state outlawed foreign currency, which I would find unrealistic but did happen sometimes in the Roman Empire.

Sources: Symbols of Power: 10 Coins that Changed the World