Indoor boulder climbing is pretty safe and climbing shoes are quite expansive. Are they really a necessity? Wouldn't my running shoes do the job just as well? Or even socks? Or ballet shoes..?
Yes. You definitely need climbing specific shoes (either your own or borrowed/ rented). They won't slide around on your feet like other shoes and they provide edging stability that doesn't rely exclusively in the rigidity of your foot. While you may be able to climb very easy routes with regular shoes, if you were to try routes with smaller footholds you'll find out instantaneously why climbing shoes matter.