In general colonies were considered completely independent cities and were founded as such--Syracuse, Tarentum, Cyrene, etc. owed no political power or loyalty to their founding cities. Colonies were obviously generally close to their mother cities, as they shared mutual traditions, cults, and dialect and often had close economic ties as well. But they were perfectly independent--Corcyra, for example, was a Corinthian colony but was constantly at war with her mother city (Thucydides even believed that the first naval battle in history was fought between Corcyra and Corinth, and conflict between Corcyra and Corinth shortly before the Peloponnesian War helped provide a major spark for the beginning of the conflict).

Special types of colonies, in which settlers retained citizenship in their mother cities and where the mother city had political and legal powers within the daughter city did exist, but are the (rather rare) exception, and far from the rule. The Athenians set up cleruchies, a uniquely Athenian imperial institution, in many of the rebellious or frontier regions of their empire. Cleruchs were given a land grant by lot, and thus became eligible for elevation to the hoplite class, which required land holdings. The cleruchies were therefore essentially military garrisons of a permanent type, and could be relied on to police surrounding areas--they were common along the Thracian coast (though possibly not Amphipolis) and in areas that had rebelled. Often critical cities that had been seized in war were settled by cleruchs (e.g. Aegina, whose citizens were expelled in 431 and the city settled as a cleruchy). But obviously this was the exception--colonists set out to found new cities, and cleruchies were not properly colonies, as they were politically dependent on Athens and often weren't particularly self-sustaining, as seen by their retention of Athenian citizenship