First, let us convert $ 629_{10} $ to each of the two bases. To convert to base 7, we realize $ 7^{4}>629_{10}>7^{3} $. So, we can tell that $ 629_{10} $ in base seven will have four digits. $ 7^{3}=343 $, which can go into 629 only one time at most, leaving $ 629-1\cdot343 = 286 $ for the next three digits. $ 7^{2}=49 $ goes into 286 five times at most, leaving us with $ 286-5\cdot49 = 41 $. Then, $ 7^{1}=7 $ goes into 41 five times at most, leaving $ 41-5\cdot7 = 6 $ for the ones digit. All together, the base seven equivalent of $ 629_{10} $ is $ 1556_{7} $. To convert to base 8, we realize similarly that $ 8^{4}>629_{10}>8^{3} $. So, we can tell that $ 629_{10} $ in base eight will have four digits. $ 8^{3}=512 $, which can go into 629 only one time at most, leaving $ 629-1\cdot512 = 117 $ for the next three digits. $ 8^{2}=64 $ goes into 117 one time at most, leaving us with $ 117-1\cdot64 = 53 $. Then, $ 8^{1}=8 $ goes into 53 six times at most, leaving $ 53-6\cdot8 = 5 $ for the ones digit. All together, the base eight equivalent of $ 629_{10} $ is $ 1165_{8} $. Finally, comparing $ 1556_{7} $ and $ 1165_{8} $, we find that digits 1, 5, and 6 are present in both numbers. Thus, there are $ \boxed{3} $ digits in common.