I signed up for the card using a relative’s UK address and received it within a week. Wasn’t asked for any KYC documents at the time. Was even able to use it a few times overseas (outside the EEA). However, Curve eventually locked my account and asked for evidence of residence in the EEA (passport, visa, recent utility bill in your name…). So they may not ask for KYC documents at signup, but they likely will eventually.