'We have colds.' 'We both have colds.' 'We've got colds.' All versions have the same meanings.
You can have similar constructions with 'headaches,' 'toothaches,' 'coughs,' and 'tummy aches,' as well.
For 'temperatures,' for the plural, I would only use it with the construction 'We both have temperatures,' the other constructions are probably fine too, but that one sounds more natural. We often say 'fever' as well-- or, 'We both have fevers.'
If you put 'a' in front (especially with the 'both' variant), then it switches back to the singular, i.e. 'We both have a cold (headache, fever, cough).'