It would not work.
There is apparently a common misconception in Georgia that this would be the case, based on Article IX, Section II, Paragraph III (b)(1) of the Georgia constitution, which says:
No county may exercise any of the powers listed in subparagraph (a) of this Paragraph [including police protection] or provide any service listed therein inside the boundaries of any municipality or any other county except by contract with the municipality or county affected.
What many people miss is the clause right before that: "Unless otherwise provided by law." Georgia courts have held that the law does provide otherwise when pursuing someone for a traffic offense:
The plaintiff contends that when the collision occurred, the policeman-deputy sheriff had no authority to be pursuing the Mitchell car because he was outside the county in which he had a power of arrest. While ordinarily a peace officer has power of arrest only in the territory of the governmental unit by which he was appointed, there are two exceptions to the rule present in this case. Code Ann. s 92A-509, which deals with arrests for traffic offenses, provides by implication that certain officers (including deputy sheriffs) have arrest powers for these offenses outside their appointed territories.
City of Winterville v. Strickland, 127 Ga. App. 716, 718, 194 S.E.2d 623, 625 (1972).
What that case decided in 1972, the principle was in place well before the boys began their hijinks.
I don't know of any state where the law is different, though the answer would be different if the boys crossed into another state.