Whether people would tell each other 'they'd been shot' is a bit complicated to answer, partly because most of our records of people being shot by arrows don't involve dialogue (unless there is a prose source, which I don't look at much, we're looking at medical treatises, chronicles and court records for our evidence which are all in fairly formal, impersonal language) and partly because of how translation works -- the original Latin may or may not say 'shot by an arrow' but as that is how we would render that sort of phrase today, it potentially has a tendency to influence translators towards using 'shot' over other phrases. Translation is hard trying to render a word from one language into a different language, when meanings don’t always match up exactly, and sometimes using the modern phrasing reads better than the most literal, direct translation.

Looking at the translated court records, 'shot' is the term used in English. For example, you get the 1298 record in the deBanco Roll reading "John fitz Simon feloniously as a felon shot the aforesaid Simon de Skeftington with a bow and barbed arrow". The original Latin reads as "Johannes filius Simonis felonice ut felo sagittavit predictum Simonem de Skeftington de uno arcu et una sagitta barbata," the important phrase being "sagittavit" or the third-person singular perfect active indicative of “sagittō”, meaning 'shot with arrows'. So it's not exactly 'shot' in the modern projectile-non-specific case, as it does specify the projectile type at the same time of saying 'shot'.

Other accounts also use 'struck by an arrow,' for example John Bradmore's Philomena, where he describes his treatment of Prince Henry's wound after the Battle of Shrewsbury: "Prince of Wales, duke of Aquitaine and Lancaster, was at the battle of Shrewsbury struck in the face with an arrow beside the nose on the left side" or in Latin, “filius et heres dicti iamscripti i/lustrissimi Regis et princeps Wallie dux acquetanie et lancastrie apud bellum de Shrouesbury in facie iuxta nasum ex sinistra parte cum sagitta fuit percussus” with “fuit percussus” forming the “struck” part.

So, saying someone had been 'shot' is definitely a possible rendering into English of what they would have said in Latin, but they did use other versions such as 'struck' and a comparison of the frequency of each would be interesting, but would really need all the sources to be in Latin.

Also on top of that, the common speech would be in Middle English, not Latin so there is not even a guarantee that the phrasing used in these very formal documents has any correlation with what people would tell each other.