How did classic masters like Alekhine and Anderssen practice tactics? Surely there was no chess.com or chesstempo at that time. How did the masters back then manage to increase their tactical awareness?
In Joosep Grents's wonderful biography on Paul Keres, there is a brief first-hand account of tactics training in that time period. For background, Paul was a contemporary of Alekhine and almost played him for the World Champion title, but the match never materialized.
Here's what Paul Keres had to say about training tactics in the 1930s (emphasis mine):
I used correspondence games, above all, for experimental purposes, often opting for very risky opening lines and attempting, at all cost, to create complicated positions which I could use to train my tactical skills... It is easy to notice a bias toward tactics in these games, while there are still considerable problems apparent in solving strictly technical problems.
The article notes that at one time Paul was playing more than 100 correspondence games at once.
Alekhine died in 1946. I have a copy of Fred Reinfeld's "1001 Winning Chess Sacrifices and Combinations" in 1955, 9 years after his death, which is basically a tactics training book. Reinfeld was the first dedicated chess author, as far as I know, who churned out chess books on an industrial scale. It could be that he was the first person to come up with the idea of extracting tactical motifs from games and putting a large number of them in a book and selling it.
Before Reinfeld there was the idea of games collections of short games which would contain tactical ideas, just not in such a concentrated form. I have a book called "200 Miniature Games of Chess" by J. du Mont which was published in 1941, within Alekhine's lifetime.
I suspect that players of Alekhine's generation and earlier honed their tactics by playing through short, tactical games, possibly from games collections like this.
Note that tactical motifs, like pins, discovered check, the importance and use of the 7th and 8th ranks and others, were identified and discussed by Nimzowitsch in his book "My System" published in 1929. These ideas were not new. What was new at the time was classifying and describing them as Nimzowitsch did.