In 1st person (sometimes, 2nd person, too) it sounds more or less neutral for some verbs like хотеть, мочь, думать. However, Russian is in general less prone to drop pronouns than Spanish, so omitting them would sound distinctly informal or confusing. I think it depends on how frequent such verb is in standard situations of social interaction or expressing your opinion—in layman's terms, stock phrases let you be less wordy and still sound natural.
In this course, we use some verbs and predicate words like that. Pay attention and everything's going to be fine :).
I've noticed it's mostly when you use хотеть and мочь ,i.e. you can say "mogu" instead of "Ya mogu" and "Hochu" instead of "Ya hochu" etc.
I haven't noticed any other case where Ruskies tend to drop pronouns, although that might be because I haven't been paying attention :-)
EDIT: Just remembered it's also true for imperatives as well, although that one is quite common with all of the European languages.