[did visitors bring servants with them?]

I found the following information in Venetia Murray's book "High Society in the Regency Period" which has a chapter on The Country House:

"It was taken for granted amongst the upper classes that a large part of the year would be spent in travelling aroud the country visiting friends and relations; and that they would be welcome at short notice, even though the party often consisted of the whole family and a retinue of servants, lady's maid, valet, nanny, governess, secretary and the ubiquitous family doctor."

She is describing the top echelon of society here - I can't imagine the Bingleys, for example had that kind of entourage and Darcy as has already been pointed out was a single man. But maybe scaled down a bit...?

I also found a description of life at Godmersham (the house where JA's brother Edward lived) in Claire Tomalin's biography of JA that says:
"The house, airy and spacious, was made for guests. It was maintained by dozens of servants. Fires were lit in all the bedrooms: breakfast was at ten, and dinner might be as late as six thirty; trays of food and drink would be brought to suit everyone's appetite in the hours between." So even there the staff was enormous.

I believe in many big houses estate workers and their families would be called in to give extra help in the kitchens etc when required.