If you each contribute separate parts (for example, one writes the lyrics while the other writes the music), you would usually each own copyright in your part. If you write music together, then you may be joint owners of copyright in that music. This means that anyone who wants to use the work would need permission from both joint owners. It is always advisable to have an agreement clarifying copyright ownership in these circumstances. This helps avoid confusion and disputes.
The best way to present your songs is to make a demo recording saved digitally. Don’t worry about the production too much but try to focus on the melody line and lyrics.