A wild guess, but there’s no need for an antenna to be a solid sheet; it could have holes or be a mesh. And it’s also probably in your interest to make your satellite invisible. I don’t suppose there are too many design documents for spy satellites around, though.
A baseball diamond isn’t that large. IANARS, but to see a baseball diamond-sized object with the naked eye is damn near impossible. I think it’s commonly told that the Great Wall can be seen from orbit, but I believe that’s been debunked.
>> Shouldn’t these things be the brightest things in the night sky?
The ISS is quite a bit bigger than a baseball diamond and it is not visible to the naked eye during the day or during the night. It is only visible at dawn or dusk when it is hit by sunlight. In any case, whether the spy satelite is visible to the naked eye or not it makes no difference to its usefulness.
Needless to say, the spy satellite is no doubt quite large as satellites go. Anything that needs a Titan 4B to get into geostationary orbit is “huge”. Just a guess, but I would imagine that our signal intelligence satellites look sorta like standard communication sats but a lot bigger and with a lot more and bigger antennas sticking off of them. A standard comms sat is probably on the order of not much more than a ton in weight. According to the Titan 4B page at Encyclopedia Astronautica it can lift about 6,000 kg into geostationary orbit.