It varied greatly between regions and, to a lesser degree, by time.

In the south of the peninsula, the de Hautevilles had once merely been the foremost family in a collection of (twelve) Norman bands. The latter were highly successful in carving out their own fiefdoms in Apulia, Lucania and Campania. While acknowledging Robert Guiscard's ducal title, they valued their autonomy and weren't hesitant about taking up arms. Just working off the index of GA Loud's Age of Robert Guiscard, there were revolts in 1064-5, 1067-8, 1072-3, 1078-9, 1079-80 and 1082-3. Partly this was due to Guiscard's failings as a statesman but it also reflects the degree to which these Norman families had established themselves on the peninsula.

Things only got worse under Roger Borsa and William II. If Guiscard had at least been able to control his subjects, with effort, then by the early 12th C, according to Loud, "most of the Apulian coast [and] the south of that province... had escaped from effective ducal authority." The arrival of Roger II to power restored some order but he had to establish himself by force and the mainland Normans continued to periodically provide his successors with trouble.

Sicily was different. It's conquest had been the enterprise of Roger I (with support from his brother) and the Norman rule of the island took on a more centralised form. There was the odd revolt by unhappy/ambitious barons (eg 1155) but by and large the island's nobles never had the strength of the those on the peninsula. Nor was there any doubting that the focus of power on the island lay firmly with the court at Palermo.

So overall the Normans on the mainland were more powerful and prone to revolt than those in Sicily. However, aside from odd periods of extended revolt or weak central authority, they were rarely autonomous enough to be considered de facto independent.