I have never owned professional dance shoes, though I am trying to purchase some now.
You'll find when you start out, it's best to wear your most comfortable heels. For me, that was a pair of 1 and a half inch, closed toe shoes with inch wide heels, plastic soles, and insoles. However, as you get better, tou can move on to higher shoes (I got up to a two inch heel, next). Eventually, you will be comfortable in what used to look very unsteady. My favorite dance shoes right now are 3 and a half inch stilletto heels, and they tie like roman sandals. I have only been dancing for one and a half years!
No you do not "need" dance shoes, though they really do help in the long run. If you really like dancing, investing in a good pair would be a wise decision. Untill then you can just use regular dress shoes, and maybe have suede applied to the bottom. One lady I know of glues felt on the bottom of her shoes!
Have fun dancing!
Kelly
PS Don't limit yourself to latin/swing. Try a little of everything!