Can one get by with a pair of regular shoes or does the water not drain well on the sidewalks that you must have at least ankle length waterproof boots if one doesn't want soaked socks? Are there puddles everywhere?
I live in Paris and I don't even own boots. I wear “normal” shoes all year round, and so do most locals (except possibly during the handful of snow days).
You do need closed shoes (i.e. not sandals) that can withstand rain in order not to get your feet wet. Paris has no dry season, only seasons during which it doesn't rain for a few days at a time. But even during the worst downpour we have here, you won't have to wade in deep puddles unless you go out in the country. More generally take a light waterproof coat with you on your trip even in the summer, but leave it where you're staying if the weather forecast says the day will be sunny throughout.
If you want to keep your legs relatively dry, the main thing to watch out for is not to get too close to the street when it's been raining, as cars zooming past can splash water pretty high sometimes (high enough that boots wouldn't help anyway).
No, you do not need rain boots in June in Paris.
First of all, even in the worst weather you can expect to walk in good shoes without problems.
Secondly, in June the chances are pretty high for warm weather with little rain, even when it rains it is not as likely to be downpours, more likely light rain.
Even when there are downpours, and/or very high water levels in the rivers, Paris has so many roads and bridges that you are very likely able to take a different route to avoid walking in the few streets that are wet. Or use public transport to avoid the problem areas.
I was in Paris in March and April this year, one time in the rain, the other time nice weather. Both times some streets were rather wet but neither time the water stayed. The streets were wet because they clean the streets with water that is pumped onto them just for that reason. There is a separate water system for that reason that uses river water that is not cleaned to human consumption level.