For low speeds, such as lower than Mach 0.5, the drag coefficient of an aircraft (or some other flying object) can be treated as if constant. Thus, the drag force is proportional to the square of the speed. If it takes an engine with a thrust of X to fly at Mach 0.1, then it takes a thrust of 4X to fly at Mach 0.2, and 16X to fly at Mach 0.4. If the drag coefficient was constant for higher speeds, then 100X would allow flight at the speed of sound (Mach 1). Noting that the power required to produce that thrust is proportional to the speed, then it would take an engine with 1000 times the power required for flight at Mach 0.1 to achieve Mach 1 (assuming constant drag coefficient). Obviously, there will be some technical challenges needed to achieve high speed flight.

How close was the challenge to being overcome in, say, 1940? Consider a fast 1940 fighter: the Spitfire Mk V. Maximum speed in level flight of about Mach 0.5, with an engine delivering a power of 1000kW, and a thrust of about 6kN at Mach 0.5. With constant drag coefficient, an engine of power 8000kW would have sufficed for flight at Mach 1, or a thrust of 24kN. By the end of WW2, such engines were available: the jet engines in the Gloster Meteor delivered 16kN of thrust each, for a total of 32kN. The Meteor, a little larger than the Spitfire, could achieve a mere Mach 0.82 with this thrust.

The problem is that as the speed approaches Mach 1, the drag coefficient increases, and the thrust required for flight at the speed of sound is much greater than one would expect if one assumed that the drag coefficient was constant. How much higher does the drag coefficient get? It depends on the shape of the flying object (and designing aircraft to reduce this increase was important for making supersonic flight routine). For objects such as bullets, an increase of approximately 3 is typical - if we can treat our Spitfire similarly, it would require approximately 80kN of thrust to achieve the speed of sound, rather more than readily available at the end of the war. This was a genuine difficulty.

Low speed drag was reasonably understood. Plenty of measurements existed, up to speeds of about Mach 0.8-0.9. High speed drag was measured too - there was no thought that supersonic motion was impossible to achieve, as it was commonly achieved by bullets, artillery shells, and the like (and rockets, such the the V2). What was unknown - lacking both good measurements and good theory to use in the absence of measurements - was how drag behaved at close to the speed of sound.

For a discussion of engineering and scientific thought on this transonic gap in measurements and theory, and the problems it posed, see [1]. The gap itself is nicely illustrated by https://history.nasa.gov/SP-4219/4219-082.jpg

So, it was known that

1. the power and thrust required to achieve speeds very close to the speed of sound increased rapidly,

2. there were additional problems with vibration, etc., that presented additional difficulties for aircraft near the speed of sound, and

3. no way of overcoming these, for a manned aircraft in controlled flight, had yet been achieved by the end of WW2.

Thus, "difficult" is clearly correct. As for "impossible", it's possible that some engineers felt it was impossible, without some major advances in science or technology. "Impossible" did appear in the popular press, and many people probably thought of the "sound barrier" as an impossible-to-break barrier.

As mentioned above, supersonic speeds were, in themselves, routinely achieved, so the "barrier" needs to be understood in context: an aircraft, starting at subsonic speeds, will experience difficulty in trying to accelerate to supersonic speeds.

It's very clear that many engineers and scientists felt that while it was difficult, it was certainly possible, and they happily continued to work on the problem. With the achievement of supersonic flight in the late 1940s, their judgment of "possible" was thoroughly vindicated.

Reference:

1. John D. Anderson, Jr., "Research in Supersonic Flight and the Breaking of the Sound Barrier", chapter 3, pp 59-89, in Pamela E. Mack (ed), FROM ENGINEERING SCIENCE TO BIG SCIENCE: The NACA and NASA Collier Trophy Research Project Winners, The NASA History Series NASA SP-4219, NASA, 1998

https://history.nasa.gov/SP-4219/Chapter3.html

https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20000012419.pdf