I think the general rule is that you always need a subject in English, regardless of whether there is a full stop. This applies to the indicative mood. The imperative is different. For example, in "Stop doing that and come over here" the second independent clause correctly lacks a subject pronoun.
Perhaps you are being influenced in your thinking by Spanish constructs such as "No tenemos nada que hacer y estamos aburridos." But as you know, in English it would be "We have nothing to do and we are bored." You won't go wrong by ensuring that your English clauses always have an explicitly stated subject. When in doubt, use a pronoun.
- For Teachers