If can I see for myself that they're learning & have learnt, that they've improved from where they were in past. What is the general trend, allowing for occasional plateaus and spells when something else (like social skills) may need first priority. It's not that the assessments are useless, but I'd think they're just one little bit of info.
Impressions gleaned from:
homework, things they say about what they've learnt, non-academic activities (how things like thank you letters improve), math riddles that come up in daily life, art work, academic work sent home, marking on the academic work, questions they ask, observations they make, etc. Also how much enthusiasm they muster for learning even more.
And even silly things like what ability group they seem to be in; obviously I worry more about progress if they're near the bottom than near the top.
What the teachers say about them; you can tell when a teacher is frustrated or delighted by a child's effort & progress, often without teacher having commented directly on either.
By now I've got DC4 at school so I've a pretty good idea what typical progress looks like in lots of areas. Experience helps.