In 2008 I was diagnosed with a bicuspid aortic valve with mild regurgitation after a heart murmur was found during a routine physical. I went through some tests with a cardiologist and got the clearance to continue training as long as I wasn't having any symptoms. I had annual checkups with echos and the occasional stress test.
Since then I've done Ironman a couple of times and have run about a half dozen marathons and a bunch of other races. I didn't notice any adverse effects on training and my race times were getting progressively faster. Last fall I hit PRs in the half marathon (1:16) and full (2:48) while running 50-70 miles a week with some hard sessions thrown in there.
In January I went to see a new cardiologist since I hadn't been in two years (wasn't comfortable with my old cardiologist). I had just started upping the miles and speed work again for a spring race and was feeling great. They performed an echo and found that the leakage / regurgitation had gotten severe and that my left ventricle was severely enlarged.
Fast forward to now and I'm about 9 weeks post surgery for valve replacement. My doctors have cleared me to run again (no hard training for now) so for the past couple of weeks I've been doing 3-4 miles a few days a week at a very slow pace. I've been told I shouldn't have any restrictions once the ventricle returns to a more safe size.
So to answer your question I was permitted to train hard and continued to do so and didn't notice any effect on my training or race times. That being said I can't help but wonder whether or not running and training led to my valve crapping out at age 31. My doctors never said that specifically so I'm just speculating. When first diagnosed I was told I would likely need surgery at some point but the timing would be different for everyone. I don't regret training and racing and hope I'm able to get back into it but like I said I can't help but wonder.
Some words of advice would be to stay diligent and keep up with your annual / semi annual checkups with your cardiologist. Just because you're not having any symptoms like I wasn't doesn't mean something isn't wrong. If your doctor doesn't give you any restrictions then I'd say train as hard as you'd like and keep it up.
Best of luck!