Anesthetic isn't used during a typical cleaning, although some patients have "sensitive areas" that a topical anesthetic may be applied to desensitize the tissue. If by "a little beyond normal cleaning" you mean a "root planning," then yes, anesthetic is used and needed for this procedure, and this is usually done by quadrants or arches due to the later discomfort. Occasionally there are patients that do not want any type of anesthetic for any procedure. We have some that have crown prep's, finish root canals or small fillings completed without any anesthetic. I'm not saying this is common at all though. We will always advise you "when" a procedure can be preformed without any anesthetic as sometimes the injection site causes more soreness than the procedure will produce. We know which will be uncomfortable and which will be painful, so it's always best to take our advice on this one. I would just take your dentist advice as he knows if your procedure will be painful to you and is only looking out for your comfort. Hope I've been of some help with this and good luck!
Additional information: Two years is fine, you shouldn't be in any discomfort with a cleaning.
work in the dental field