Hey ProfQuelaris!
Your question seems basic, but to answer it two more questions first need to be addressed: when and where. I won't go into all of the detail, that would be too long, but here's a basic summary.
First, let my start be clarifying that Quenya and Sindarin are two related but nonetheless distinct and non-mutually intelligible languages. They arise from a common ancestor, called Common Eldarin (CE for short). There's a big time gap between CE and Quenya/Sindarin, so for the when we can at least assume that the earliest historical point we are looking into is after the two languages have already formed into their (relatively) modern form. Shortly before the destruction of the Two Trees seems like a good starting point. (Note: If you find some of the references to the lore confusing, you can consult the summaries on Tolkien Gateway.)
With this in mind, let's talk about where. Elves are not a single people: multiple Elvish kindreds/civilisations can be found around Arda. Some of them speak neither Quenya nor Sindarin, but other Elvish languages - let's put those aside. At the time of the destruction of the Two Trees, Sindarin was spoken by the Sindar in Beleriand, whereas Quenya was spoken by the Ñoldor and the Vanyar in Valinor. At this point both languages are the every-day languages of their respective peoples.
When a part of the Ñoldor returns in exile to Middle-earth, as described in The Silmarillion, things start getting mixed up. The Ñoldor are still Quenya speakers, but they quickly learn the local Sindarin language to communicate with the local Elves. Thingol, king of the Sindar, then hears of the Kinslaying of Alqualonde, and bans the use of Quenya in his realm (= all of Beleriand). This event ultimately leads to Quenya no longer being used as an every day language by the Ñoldor in Middle-earth in future ages. They do, however, retain it as a language of lore and literature - indeed somewhat like Latin (though I should note that Quenya is more strongly inspired by Finnish).
But here it must be noted that Quenya did not in fact die out as an every-day language - the Vanyar and those Ñoldor who remained in Valinor still use it!
In the Second Age Sindarin rose to further prominence, for two reasons.
First, the Edain who now settled on Númenor carried both Sindarin and Quenya with them, alongside their own languages. They too used Quenya only for lore, and Sindarin was a bit more widespread (though the Adûnaic tongue prevailed in any case), but the important point is that once the Númenóreans started sailing to Middle-earth, Sindarin was the common language they shared with the Elves there, so it was a useful language for communication.
Second, the Sindar and the Sindarin speaking Ñoldor were forced to move away from the sunken Beleriand at the end of the 1st Age. Many of them settled in Lindon, the remnants of the easternmost part of Beleriand that was once known as Ossiriand, and which used to be the home of the Green Elves, who previously spoke a different language (a relative of Sindarin). But many of them moved even further east, into Eriador and even beyond the Misty Mountains, where Nandorin Elves lived. In fact, the Nandorin populations in Lothlórien and Eryn Lasgalen (=Greenwood the Great, later known as Mirkwood) eventually came under the rule of Ñoldor (Galadriel in Lothlórien) and Sindar (Oropher and later Thranduil in Mirkwood). Consequently, though other Elvish languages still survived, Sindarin became the main every-day language spoken by all Elves in the west of Middle-earth.
When the Númenóreans settled the kingdoms of Gondor and Arnor in the late 2nd Age, Sindarin was basically solidified as a lingua franca used by Men and Elves in the west of Middle-earth (though I daresay Mannish and Elvish Sindarin were somewhat different dialects at this point).
Throughout the 3rd Age the use of Sindarin amongst Men declined, and the population of Elves in general declined as well, but Sindarin nonetheless remained the central every day languages of the Elves, whilst Quenya was left as a language of lore and poetry amongst the Ñoldor (in Middle-earth). There is definitely no guarantee that the average Sinda would know much or any Quenya, by the way.
I hope this answers your question somewhat!