This is a common question I get from students learning to fingerpick. The short answer is you do not need to have nails to do fingerpicking. Now for the longer answer 🙂
When you fingerpick without nails the sound is softer and less distinct. What I mean is that the individual notes are not as clear. When you use nails it adds a little bit of bite to each note and makes it clearer and more distinct. To some degree it depends on the style of music you’re playing, but generally I prefer the sound with fingernails.
Now to be clear I’m not actually picking just with my nails. I’m hitting the string with the flesh of my finger first and then the string to rolls off the tip of my nail. In essence I’m using both the fleshy part of my finger and the nail to get the sound I want. If I use just the nail the sound would be very tinny and not very pleasant.
To do this properly not only takes lots of practice but you’ll need to shape and smooth your nails regularly. I recommend if you’re just starting out to practice without nails. As time goes on you can decide if you want to grow your nails and shape them in the appropriate way for fingerpicking.
As a side note: if you want to play either classical or flamenco guitar properly you would definitely need to grow your nails and learn to shape them properly. It is not uncommon for a good classical guitar teacher to spend an entire session or two on shaping your nails properly.