this is very easy to accompish.
First you need a source video for your snow. For example like this one.
download it, then in Premiere place it above your other video and in >effect controls > opacity, play around with the "blend mode" settings and chose one that suits your taste.
it works the same way with the circles. you just need to use a different source video.
If you want to create the source video by yourself, then you should use After Effects. Do a search for the plugin "particular" - by Red Giant.
or search for "create snow in red giant particular"
here's a quick example: