Summary: There are studies showing a difference in gastrointestinal problems and studies where no difference was observed. At least some of the effect it has on the stomach is not related to whether naproxen is released in the stomach or small intestine.
There are two very small studies here:
that concluded that the negative gastrointestinal effects were lessened in the coated form. However, they don't disappear, pointing towards there being both a systemic and local effect of naproxen for the stomach, independent of how it's delivered.
A slightly larger (though still small) study The efficacy and tolerability of enteric and non-enteric coated naproxen tablets: a double-blind study in patients with osteoarthritis found no significant differences in either efficiency or tolerability, meaning that gastrointestinal problems appeared for both forms.
An overview by Roche Pharmaceuticals contains this passage:
[...] These studies indicated that EC
NAPROSYN and NAPROSYN showed no significant differences in efficacy
or safety and had similar prevalence of minor GI complaints. Individual
patients, however, may find one formulation preferable to the other.
Five hundred and fifty-three patients received EC-NAPROSYN during long
term open-label trials (mean length of treatment was 159 days). The rates for
clinically-diagnosed peptic ulcers and GI bleeds were similar to what has been
historically reported for long-term NSAID use.
However, A double blind study comparing the efficacy and safety of enteric coated naproxen to naproxen in the management of NSAID intolerant patients with rheumatoid arthritis and osteoarthritis. Naproxen EC Study Group found a significant reduction of gastrointestinal complaints by at least 15 percent. Again, efficiency was the same.
To me, that looks like it's at least worth taking the EC form, since they both work equally well and the EC form might have an advantage.