The concept of the four fountains decorating the recesses of the buildings was preferred over the original idea of the Pope who commissioned the job, Sixtus V, that of a fountain in the middle of the road to celebrate the conclusion of the Alexandrine acqueduct.
The crossroad, formed by the two roads originally called Felice Street and Pia Street (the names would be changed after 1870, with the dissolution of the Papal States), was the result of the works on the aqueducts, and it deserved a proper decoration.
The travertino marble used for the statues comes directly from the Septizodium, a facade decorating a nymphaeum located the bottom of the Domus Severiana, on the Palatine Hill.
This is the first mystery of the fountains: nobody knows who are their designers or their sculptors.
There are a good number of popular names sprayed around the web, but truth is that none of them can be ascertained.
Some of them are errouneusly attributed to Domenico Fontana, but he actually just designed the original Felice Street (the modern Quattro Fontane Street).
The lack of information in this sense depends probably on the fact that the fountains were commissioned by private citizens who owned the close-by buildings or the lands around the crossroad: as they weren't meant as the usual form of Reinassance propaganda, there was no need to celebrate the artists and the families behind them.
The fountains have been recently restored by a private contribution of Prada, so I profitted to take a look and some pictures (you know, before some soccer enthusiast from Northern Europe decides to blow them up).
It's generally accepted that the two male figures represent two rivers, identified as the Tiber (because of the presence of the She-wolf, even if according to certain sources that was a later addition) and the Arno (because of the Marzocco), and the two female figures are two Roman goddesses, Juno, representing the Strenght (because of the lion and the crown in her hand) and Diana, representing the Chastity (because of the dog next to her).
Yet, the symbology of the statues is still pretty much unclear. The statues of the Tiber and Juno sport two beautiful trees, an oak (Strenght) and a palm (Victory), but there seems to be no relationship with the two figures. There is a swan next to Juno (some identified it as a peacock, an attribute to the goddess but-- C'mon, that's not a peacock!) : as it can be interpreted as an alias of Jupiter, that would make little sense as that would be associated with Leda.
Under Tiber's and, you can spot a snake: another mysterious symbol.
Compared to the other two statues, on top of it, the Arno and Diana look overly simplified, but that can be a result of the private commission.
Also, the fountains were actually built at different times, so this may be another discriminating factor.
I'm also perplexed by the fact that this crossroad doesn't sport one of the typical Madonnella, pretty much an urbanistic standard at the time.