I don't have a profile picture, for privacy reasons. Anyone on this site could see it & I just feel better not showing it. I am a very trustworthy guest, and you would really be losing out if you were to follow the advice to decline bookings en masse just because someone hasn't disclosed a ton of information for public consumption.
Usually when I request a booking, I explain that I am happy to send pictures of myself and my husband, as well as provide additional information about my background. Sometimes I forget, with all the back & forth, but I do try to talk about myself when I request the booking.
I suggest sending a polite message asking the guest to provide more information (maybe specifics about what you would like to know.) I am happy to send a picture once I'm booked, so that the hosts can confirm our identities, anyone who refused that would be unreasonable. If the person is unresponsive, since you are in a time crunch to approve the booking, ask the person to call you so you can get acquainted. Anyone with some sense will be happy to comply if he wants the booking!
Ultimately it is the responsibility of the guest to make sure the host is comfortable having them in his or her home.