Does local anesthesia make you high if it gets in your blood?

Dizzy . It may make you feel dizzy. May cause ringing in ears or funny taste in mouth. Will not make you high.
No. But the Epinephrine (adrenaline) in many local anesthetic solutions may make your heart race briefly or give panicy feelings that go away quickly. Toxic doses of local anesthetics can give central nervous system side effects, but that will not occur if your surgeon, anesthetist is using appropriate doses.
No. If local anesthesia gets in the blood stream some patients experience very short term numbness or tingling of their face lips and tongue. If the local anesthetic has Epinephrine in it, it may cause a temporary speeding up of your heart rate. Depending on which local anesthetic is used, injecting directly into the blood stream can be extremely dangerous, fatal even. There is no "high"
No. If it contains epinephrine, direct injection of local anesthesia into a blood vessel can make you heart race for a while and make you feel dizzy, but it will not make you feel high.
Not really. A local anesthetic in your blood stream will initially have central nervous system effects up to and including seizures. Then cardiac effects will be noted which can lead to death. This is not a medication that should be experimented with. I hope that this answers your question.