That question really should be two questions:
What is sport-specific training?
Is it needed?
Depending on who you ask, sport-specific training can mean several things. For some it means that you have to duplicate the demands of the sport as closely as possible. Using movement patterns, speed of contraction, work to rest ratios, etc. that are similar to those found in the sport itself.
For others it simply means including whole body movements and explosive exercises to improve power and speed.
I tend to side with the later.
I personally do not like trying to copy the demands of the sport too closely, especially when it comes to exercise selection. I've seen many coaches go a few steps too far, taking perfectly good exercises and throwing them out of their program because they were not specific enough, or inventing dumb exercises in hope of copying actual sport movement. In reality, Siff and Verkhoshansky have demonstrated that strength training exercises that are too close to an actual sport movement can actually be detrimental.
Now, if you go with my definition of sport-specific training: emphasizing movements that include a lot of muscles at the same time as well as exercises to build overall strength (e.g. deadlift, squats, push press, bench press, chin-ups), power (olympic lift variations, loaded jumps, throws) and speed (jumps, agility drills, sprints) you can ask the question: is it needed?
After having trained over 500 athletes from 27 different sports, as well as competitive bodybuilders or the highest level, I would say yes, with a "but".
The vast majority of people that I have trained as bodybuilders and who were not naturally gifted athlete stayed below average athletes even if they got really strong and muscular. One of them, an IFBB pro, can't even sprint properly! The only bodybuilders who i've seen have any resemblance of athletic ability were those who played competitive sports before doing bodybuilding or those who were naturally gifted athletes. All the others were below average in any activity requiring speed, agility, coordination and endurance. That doesn't mean that bodybuilding training made them bad athletes, but it certainly didn't help.
Now, if someone is a naturally gifted athlete, fast, explosive, agile. Chances are that he will not "need" sport specific training as much as others less gifted individuals because he already has many of the needed tools. A person like that will likely benefit from the most general training simply by becoming stronger and more muscular. Until a certain point that is.
But the further away you are from being gifted, the more you will need to use exercises to improve explosiveness, speed and agility.
Another factor to consider is the amount of track work one is doing. For example if an athlete is doing sprints and agility drills 3x per week he will likely not be as much penalized from not doing sport-specific strength training because he is building speed and explosiveness already and just getting stronger might be enough.
With the football and hockey players I train and have trained I noticed a strong correlation between sprinting speed and the power clean test. I noticed a similar correlation between the vertical jump test and the power clean. But the correlation between the speed and jumping tests and strength lifts like the squat and deadlift was much lower.
IWhen I played football I trained a lot like a bodybuilder and ran a 5.32 40yds and had a 28" vertical. When I trained and competed as an olympic lifter I tested at 4.54 (electric time) on 40yds and tested at 42" on the vertical jump. If I had trained like that as a football player, there is no doubt that I would have been a much better player.