The standard way to call a contra dance now is prompting. Just before each figure should be done the caller says it's name using two to four beats and a minimum of words. Seth Tepfer gives a good example of this in his Great Dances for Novice Caller's Handout:

Small Potatoes -- Jim Kitch -- duple improper

A1	(16)	Neighbor balance and swing
A2	(8)	Circle L
	(8)	Ladies chain
B1	(8)	Ladies dosido
	(8)	Partner swing
B2	(8)	Circle L 3/4
	(8)	Next neighbor dosido

Would be called something like (bolds are downbeats):

........1........	........2........	........3........	........4........	........5........	........6........	........7........	........8........
-	-	-	-	With your	Neighbor	Balance and	Swing
-	-	-	-	-	-	-	-
-	-	-	-	Join	Hands	Circle	Left
-	-	-	-	Ladies	Chain A	cross the	set
-	-	-	-	Ladies	Only	Do si	Do
-	-	-	-	With your	Partner	Swing	Now
-	-	-	-	Circle	Left	Three	Quarters
-	-	-	-	Next	Neighbor	Do si	Do

Note that the first line of the table above is actually the last eight beats of the previous time through the music. Alternately, at the beginning of the dance, the "with your neighbor balance and swing" could be the four potatoes.

If I were actually calling it, I would use fewer words and not use four beat calls for everything. Something like:

........1........	........2........	........3........	........4........	........5........	........6........	........7........	........8........
-	-	-	-	With your	Neighbor	Balance and	Swing
-	-	-	-	-	-	-	-
-	-	-	-	-	-	Circle	Left
-	-	-	-	-	-	Ladies	Chain
-	-	-	-	-	- Ladies	Do si	Do
-	-	-	-	-	-	Partner	Swing
-	-	-	-	-	-	Circle	Left
-	-	-	-	Next	Neighbor	Do si	Do

(the unbolded 'ladies' goes after the downbeat).

Looking through Chris Page's helpful index to the online Northern Junket, I found this on page 20 of Volume 1, Number 9, from January 1950:

Calling a Contra Dance

Mr. E. N. Weber, Watertown Mass. and Richard Lee, Brooklyn N.Y. have taken the time and bother to write requesting that we give them a contra exactly as we would call it at a dance. Many others have told us the same, and here it begins to look like it ought to be done. So here goes. Get the Northern Junket, vol 1 no 6, and tuen to the contra dance: Lady Walpole's Reel. Get a record of "Reel o' Stumpie", take a deep breath and call it with us, like this:

Balance the one in front of you
Then you swing her and she'll swing you
When you've swung you leave her alone
Go down the center with your own.
Go down the center two by two
And bring your partner home with you.
Come right back to where you begun
And cast off with the one you swung,
Chain the ladies over and you chain them right back home again
Take that lady with you and promenade across the set
---- turn around and right and left back
---- now balance the NEXT in line.
You balance there and keep in time
Then give her a swing while I think of a rhyme,
---- go down the center with your own
Down the center now you'll go,
Click your heels and stub your toe,
---- cast off and the ladies chain,
You chain the ladies over
And you chain them right back home again,
Take that lady that you swung and promenade across the set,
Promenade her half way then turn around and right and left back,
Cross at the head and cross at the foot and balance the NEXT below.
...

This is a "patter" call: the caller is calling on nearly every beat of the dance. If you look above at how I would have called "Small Potatoes" you'll see that I'm talking about 1/4 of the time. Squares can take patter (though new england squares are generally phrased and prompted like contras), but I've never heard a contra called this way. The introduction makes it sound like that's what was common.

I've never heard a recording of contra dance calling from 1950, but there is a recording online from 1965. It's Duke Miller calling a night at the Peterborough Golf Club on August 20, 1965. On those contras he doesn't go to the extreme of calling on nearly every beat, but he's calling maybe 1/2 to 3/4 of the time. He also calls every time through the dance, not dropping out after two or three times through. My parents say that when they were younger (1970s) callers would often call the whole time through except for one time near the end where they would say "you're on your own" and leave the dancers to do it from memory. I don't think duke miller does that here, though I could be missing it.

These two examples of calling represent a very different aesthetic than what I'm used to. What I think of as the ideal caller is one that teaches well and efficiently, calls just as much as is needed to get the dancers going, drops out on individual calls as soon as the dancers won't need them, then gets out of the way and lets the band do their stuff. If the band wants to do anything vocal (like blue moose was doing at the 5/15 bida contra) then dropping out is especially important. This approach is totally different from the style ralph page and duke miller are using where the caller is almost part of the band, singing or chanting the calls musically. It's not the extreme of a singing square, but it's still part of the entertainment.

I wonder if this is related to the decrease in squares? The program from that night in 1965 has 14 squares and 5 contras. A dance now is surprising if it has more than two or three. Most often there won't be any. In a square you have to keep calling because the square as a whole is short (7 times through the music for a common square vs 17 for a common contra) and there is more to the dance (one 64 beat pattern for the contra, two (figure and break) 64 beat patterns for the square. Not only that but the square generally won't do the break the same way each time and often the caller will add other things. If you're mostly doing squares, and squares require calling all the way through, dancers don't learn how to deal with the caller no longer calling and the caller never learns how to drop out smoothly and gradually.

More speculation, but dropping out may be older than the decline in squares. If a dancer is also calling, it takes a lot of breath to do both. So I'd expect that economy of words would be enforced, and as soon as a call was no longer needed for helping the dancers the caller would stop giving it.