Numbness. Local anesthetics are used to temporarily block pain at the injection site, but they can spread to nearby structures as well. Dental anesthetics can cause difficulty with speech, eating, and occasionally eyelid drooping. Furthermore, numbness of the mouth can make one more prone to tongue, lip, or cheek biting, so it's important to remain aware of that possibility. Rarely infection from injection.
Rare side effects. Depends on the anesthetic. Cetacaine, benzocaine, and prilocaine can cause methemoglobin where your red blood cells cannot bind oxygen. This can cause chest pains, shortness of breath, heart attack, cardiac arrest. If local anesthetic has a high absorption in your blood vessels, this can lead to excitability, extreme anxiety, elevated heart rate or decreased heart rate, seizures, coma, and death.
Varies. Oral anesthetics are generally very safe in oral care. They may burn initially, and if numb, it may be easier to cause trauma (e g bite your tongue and not feel it). Similarly, if the throat is numb and liquids are swallowed, aspiration (fluid going down the windpipe) is possible. High doses may cause systemic side effects (dizziness, nausea, ringing in the ears, or more serious effects).