First off, did a sentry network like this exist? Was it always in operation, or only during times of perceived threat?

Coastal beacon networks in the British Isles had originated with the Romans, but had reached a high point during the Early Medieval period, when an extensive network of coastal and inland sentry points had been connected with the Burghal network during the Anglo-Danish wars of the Viking Period. These had later been updated during the reign of Edward III, who had ordered their bonfire mounds replaced with elevated pitch barrels.

By the end of the medieval period, the network appears to have lapsed largely out of use, or at the very least significantly contracted, given the effort that the Elizabethans put into restoring and expanding it. Responsibility for coastal defences rested ultimately with Sir Roger Twysden, who was Elizabeth I's Captain of Defences, although given the relative lack of centralised government and standing army, this responsibility appears to have been delegated to a county level to actually have been effected. Lord Cobham, for example, was congratulated by the Privy Council for the thoroughness with which he re-established the "ancient sea-watches" in the Isle of Sheppey and Romney Marsh.

Twysden appears to have been a fairly conscientious officer, or at least one who understood the importance of good research. He appears to have made prominent use of local historians and antiquarians in determining the locations of former beacon sites and restoring them to full use. William Lambarde's 1576 A Perambulation of Kent, for example, contains "an assesse of the ancient watche and warde of our ancestors" and notes a number of Early Medieval beacon sites that were then restored to function. Lambarde also carried out further assessments along the South Coast as far as the Isle of Wight. A handy rule of thumb for identifying sites that were in Elizabethan use is to look out for "beacon" toponyms: Although the use of here-beacen terminology is well-attested in Old English, the phrase doesn't appear to have made it into common toponyms, tot and weard being far more common. Where "beacon" exists in a toponym, it likely indicates an earlier site that was re-used in the Elizabethan period.

Keeping a standing army is expensive, so it's unlikely that professional soldiers were used to staff beacon sites. Indeed, given the phenomenal expenses incurred by English military activities in the 1540s, the majority of Elizabethan military land power remained tied remarkably closely to, essentially, county-level regimental raisings and levies (Hammer, Elizabeth's Wars). Just as the West Saxons and Mercians staffed their beacon sites with fyrdmen, so the Elizabethan sites were most likely staffed by militia. The 1558 militia statutes encouraged higher enlistment and mandated that each Lord-Lieutenant was to organise:

a convenient and sufficient number of the most able, by the reasonable charge of the inhabitants of each shire, to be armed and weaponed, and so consequently taught and trained.

One of the most important elements in the new legislation around these trained bands - apart from the direct expectation of training - was that militia enlistees were now guaranteed that they were solely for domestic use, and could not be pressed into a royal army for foreign service (Scott, The Maligned Militia). These guarantees helped to rapidly increase rates of recruitment, and meant that there was commonly a sufficient pool of manpower available to ensure that watch sites were suitably manned. Although the sites themselves may appear relatively isolated, like their earlier counterparts, they were rarely more than an hour or so's walk from the nearest settlement.