Did they ever!

Firstly, you must realize that the idea of heroism in Rome to the everyday citizen travels back to the basis of Roman Ideals. The Romans saw themselves, every day citizens, as soldiers in a kind of way.

'[The Romans] belief that monarchy always turned into tyranny was expressed in Livy's story of the rape of Lucretia... The assault [rape] of Lucretia... [lead to] her relatives and friends, led by Lucius Junius Brutus, expelled Tarquin in 509 B.C.E. They then created the republic, to ensure the sharing of power... the Romans prided themselves on living under a [free] political system.'

This story, the rape of Lucretia, which may or may not be true, shows the basis of Roman thought. They are free, ostensibly egalitarian people, who want all men to be equal, and would defend this right by force.

Furthermore, the Roman Religion/Culture aided soldiers. A shrine to the ancestors, or 'Lares', of the family in the house honored those who had committed heroic acts. It is important to realize, however, that the Romans did not see heroism as just strength on the battle field, but also as political will to stand up for Roman values. Tiberius Gracchus, father of the Empire, it is said, had a cast of his father's face that he kept at the family shrine, to honor his father's important political actions towards relieving the plebeians of Rome of financial burdens.

However, it really isn't towards the end of the Republic that we see the shift from honor as political strength and toughness to actual warrior like strength until the Punic Wars. Starting around this time, and much after, Plebeians found that joining the army was a good way to relieve financial burden. Due to so many signing up, it became a very normal thing for a father to serve in the army, and the wife and child(ren) to journey to a city to hopefully find work (most likely as a prostitute or low wage worker). The Roman men off fighting, then, would need to fight for many things. The well-being of their family (if they died, their land which their widowed wife relied on could be taken away. Obligatory: Just grab some land!.) They also fought for their ancestors, and the hopes of being honored. Last of all, and perhaps the most important, Romans had a well-known sense of 'Country First', that everything they did had an impact on their country, and they had to do their best to fight for it.

IN CONCLUSION/ TL;DR: Although Roman Toughness had always been part of their culture, it wasn't until soldiers became much more common that they were seen as an everyday part of society. Hope this answers your questions!

Sources: Hunt, et.al, 2011 version.