Some champions (like Janna's Ctrl + 2) will always trigger the same lines as there aren't any other lines in place. One of the reasons it's a Client-side thing rather than server-side is due to different languages. If there's a sassy line that's said in Polish but English doesn't have such a thing, what would take precedence? Does the English player hear Polish lines, or does the English player then lose the immersion and hear nothing while seeing a champion visibly emote (leading to a LOT of false bug reports for missing audio)? That would require a bit more resources as Quepha said earlier making extra information being sent between the client and server every single time someone triggers their emotes. It would also require you to download every language pack if you wanted to hear what they said in the native language, significantly increasing the amount of space taken up on each player's hard drive. The plus side is there can easily be an option to disable "foreign" voices for champions, the down side is there's not an easy way to detach a player from the server when that happens, so it would likely alter ping (although I'm not sure the amount of resources being tied up is significant, it's definitely something to keep in mind for coding purposes forcing players to sync up).
Aurelion Sol has a similar feature to "syncing" with his Stars that fly around him, so it's definitely possible. His Stars are given a position from all clients and registered together to form an accurate representation of where the actual location of his orbs are. This helps to prevent randomly being hit from across the map by them, or that occasional player that's "cheating" (in a similar manner to Xerath in the past few weeks) and causing his orbs to send information that he's hitting everyone all over the map. Sometimes this causes random issues (such as de-syncing) or visual bugs from the spaghetti that ensues (https://www.youtube.com/watch?v=QfJnoYTPuQ4).
For someone like French Rammus (https://www.youtube.com/watch?v=zrn2K4eFl5g) spamming emotes that has a fair amount of voice lines, it would affect the player that's playing on an English client as information would be sent from the server to trigger a voice line that isn't there. At that point it's almost like a mini-DDOS if you're able to consistently send a ton of information, although it would effectively DDOS yourself as well.