No - you do not need to enter any payment information when you sign up for a 14 days free Premium trial.
If you are asked for a payment method it is either because you are
- already subscribed to a Premium trial under the Doodle account / email you are logged in with or
- you were subscribed to a Premium trial in the past with the same Doodle account / email.
Check your account settings: www.doodle.com/settings/premium to see if you are / were already on the Premium trial.