Russia's fears were bullshit. Now before you comment, you need to read the below very carefully and read the slides or this will go way over your head. There are two types of missile defense in question: GBMD and Aegis BMD.
The ABM shield in question at the time was the Groundbased Midcourse Missile Defense System, currently in operation in Alaska and California (with a site opening in the North East some time in the next few years). It's this.
When most people talk "US Missile Defense" this is what they mean. This is "Star Wars". It's not nearly done yet, and is still flawed, it's getting there (the next step is a better interceptor in the next few years).
GBMD could defend against certain types of attack from Russia if an East Coast base is built (which it is). However basing GBMD interceptors in Europe was never a threat to Russia. Why? This is why.
Ballistic Missiles targeted at the US would never "catch up" to Russian ICBMs from European based interceptors (they would from US based of course, opposite direction). Furthermore Russia targeted Europe during the cold war with Medium, Intermediate and Short Ranged ballistic missiles. Europe was too close for
Intercontinental Ballistic Missiles to be used (for the most part).
Putting GBMD Interceptors in Europe would have done nothing to undermine Russia's deterrent. The US is going to do that anyway by the way, by building an East Coast base.
GBMD requires a large network of Radars, including an XBAND Radar to function. Russia's proposal was to base it at a Russian base in Russian territory. That was unacceptable to the US. The US has a 35 year lead on this technology, technology that Russia hasn't dropped any amount of money into. Using a joint radar site would have given them access to some of the mos sensitive US military technology there is. And it may not have even been compatible. So no plan was reached.
Later, Obama decided to replace the idea of basing GBMD in Europe with placing four Aegis Destroyers with this Ballistic Missile Defense System, around European waters performing that task, until the Aegis Ashore system could be built (basically Aegis BMD without the boat). The Donald Cook is one of those boats.
This is a picture of the one being build in Romania last year. More are on the way.
These things are very popular. The US is building a couple along the US Coast. Japan wants them. Others have expressed interests.
Basically Russia screwed up. Aegis BMD is simpler and more mature than GBMD, but less capable.
See that? It can't intercept ICBMs as well as GBMD (see footnote), but it has full capability versus Short, Medium and Intermediate Range Ballistic Missiles. In other words, exactly the kind of missiles that Russia would launch at Europe and US forces in Europe.
So that's what Russia got for it's ridiculous demand and inflexibility: a worse problem. The US sharing Ballistic Missile Defense technology is a non-starter, but the basic geometry behind how GBMD would never have hurt Russia's deterrent should have made their assent straightforward. Instead, GBMD got replaced with something that actually undermines Russia's deterrent.