No data about users is saved, this information is only retained on the server for the duration of the session and deleted on logout, some guild information is saved on the server, this includes your guild id which is stored in a database along with the sprites to determine which server the sprites belong to. The sprites you upload are saved into a MySQL database along with the guild id, this data is retained on the server until deleted by the user. This is different for premium, see here.
QuoteBot and the QuoteBot site are run on seperate servers to allow better performance, when a user in a guild runs a command that requires getting a sprite image, a request is made by QuoteBot to a private API that allows the bot to query what sprites a guild has and request to any of those sprites depending on what command has been issued by the user. The bot may also upload these sprites to discord depending on the user command (either a sprite in a generated textbox or if the user uses a showsprite command), this means by uploading a sprite to the site, you give consent for that sprite to be uploaded to discord by the bot, this also means you give permission to discord to host your content on their cdn network, see here. By uploading sprites, you have the rights to allow discord to host them.
Data you share logging in with discord or general use of the site is not shared with anyone, data you upload may be uploaded to discord, as stated in the last paragraph.
If you have any concerns, you can ask about this in the QuoteBot support server.