I've set up Beds24 for my villa site http://www.serendipity-villa.co.uk but I can't find a way of allowing bookings to overlap.
By this I mean; a guest leaves Sunday morning and another guest arrives the same Sunday evening (back to back bookings)
That is fully normal to have one booking check-out and another check-in on the same day.
Make sure you are not confusing the "Last Night" date with the "Check-Out" date (which is 1 day later)
The calendar shows the nights booked so the check-out day will generally be available for a new booking if it is not already booked.