You do not need an instrument in-hand before meeting your teacher. In most cases, especially for first-time students, your teacher will want to assist in finding the right instrument for you. This is especially true when renting an instrument, as your teacher may have preferences as to the shop you rent from. Many times, your teacher will have long standing relationships with Luthiers or Music Shops.
If your teacher trusts the person that you buy or rent your instrument from, chances are you will get a good deal on an instrument that works for your style, age and ability level. Incidentally, asking your teacher to assist in your search for an instrument, or for their approval and advice about your current instrument, shows them that you have a genuine respect for their expertise.