I am happy to let my kids (12 & 16) walk Millie. Rob (12) will happily take Millie on a pavement walk and on the long line/ stretchy lead in the park. Kate (16) will also take her to the local (large and usually quiet) recreaction field and let her off lead.
The only rules we have is that they tell me where they are going to walk and that take their mobiles and poo bags
with them. (If there was a problem I could be with them in minutes)
Rob was about 11 1/2yrs and Millie 1 year old when he started taking her for short 'round the block' walks. He did all of Millie's puppy (& clicker) training with me and whilst on walks together often had control of Millie (with me giving advise if nec). So he was well practiced before 'going solo'
They don't have to walk her every day, thats my and OH's responsibility, (but they will often come with us) but occasionally (usually in the school hols) I'll ask them to take her for a walk to help me out and I appreciate their help.
As far as a dog-on-dog incident goes (not had one yet & Millie's coming up two years) you cant stop doing everything just because 'something might happen'. If I took that attitude Millie would have alot of very boring pavement walks and never see a field, footpath or woods from one week to the next.