if you add a custom livery, can other people in multiplayer see it as well without the file?

If you upload it on the web then ye should be good

Quote from MousemanLV :

If you upload it on the web then ye should be good

what do you mean? so it only appears to people who download the livery?

First, by default, only S1/S2/S3 users can upload skins, demo users have 0 skin slots by default (demo users can buy slots for a very small amount of price, though).

If you upload the skin to the LFS web (do it on: https://www.lfs.net/account/skins), LFS will automatically handle distribution, so you don't have to worry about it. For people who meet you online, LFS will automatically download skins in the background, they don't have to download or do anything special to see your skin.

Read more: https://en.lfsmanual.net/wiki/Skin_Tutorial#How_can_other_people_see_your_skin_in_LFS