In Israel, most shuls have parsha sheets that are available for taking. There are numerous halachic problems that arise. One is that most of them have advertisements which one is not allowed to read on Shabbos [that is also important to know for those who read newspapers. Even if one is lenient and reads newspapers on Shabbos - which in my humble opinion is quite problematic and I won't elaborate here - it is forbidden to read the advertisements. It is also virtually impossible NOT to read the advertisements which jump out at you on each page. Quite often, whatever they are advertising it seems that they are really advertising גילוי עריות. That is another problem...].
Another problem is that people read these sheets instead of davening. This is ALSO a problem when people learn regular seforim instead of davening. I think that this is due to the fact that learning has the element of being intellectually stimulating while davening is plain old boring. Same story each time. Monotonous and repetitious. That is why people get excited about a hallel or when we start saying לדוד השם אורי in Elul. Finally, some VARIETY in our diet. OF COURSE, it shouldn't be this way. When one is speaking to a beautiful girl that he wants to one day be his wife he enjoys every second of the conversation even though the actual content is quite dull. I often try to listen to these boy-girl chit chats to understand what the two of them are enjoying so much. It is invariably a very dry, superficial conversation [the more interesting one's they probably have out of earshot]. So, by golly, what are they so excited about?? The answer, of course, is each other. So when we daven, we should be in love and extremely interested in our Beloved. Also, tefilla is PREGNANT [bshaah tova] with meaning. PLEASE "learn up" the siddur and find out for yourself. My personal favorite? עולת ראיה. Davening SHOULD BE the most stimulating-growth-inducing-exciting-euphoric-semi-prophetic-surreal experience of the day. HALEVAI!
Another problem is sheimos. Some of the sheets end up in the garbage. There ARE poskim who are lenient and maintain that one may wrap them up in a plastic bag and place them NEXT to the garbage bin and let the gentile sanitation worker [isn't that SO MUCH MORE REFINED than "garbage man"] take it away. Rav Shternbuch bases this on an careful reading of the Rambam who says that kisvei hakodesh [actual Tanach] must be placed in sheimos while about other Torah materials the Rambam just says to make sure not to treat them with disrespect. Others say that one may recycle the paper. Others say that one should put them into sheimos. [See the sefer Ginzei Kodesh of Rav Feindlander Shlita].
What is good about these sheets is that they increase Torah in the world. My daughter explained the psychology to me. People are hesitant to pick up a 600 page book because it is just TOO MUCH and to read a page or two feels inadequate. But if the whole sheet is only a page or two and a person reads the WHOLE THING he feels a sense of accomplishment and completion. And I see that people really read these sheets a lot more than they read seforim [during official non-learning hours such as Shabbos]. My sheet, Baruch Hashem is quite well read and it is a ZCHUUUUS to spread Torah.
What about divrei torah in newspapers? It would be better if they didn't have it because these newspapers end up in the garbage. Some say that one should cut out the Torah material and put the rest of the paper in the garbage. This is quite time consuming for papers such as the Jewish press which has a lot of Torah content interspersed throughout the paper. Others say that one may put the whole thing in sheimos. Others yet argue that this is bad for the environment. They are RIGHT. But so are disposable dishes and plastic bags and diapers and .... and.... yet people don't hesitiate to use those. So I don't know. Ask your Rabbi:).