Safety of DSA key parameters sharing
In DSA (and ECDSA), it is possible and common to share the same domain parameters, across multiples users. AFAIK (and according to common wisdom, including FIPS recommendations) this introduce no known security weakness. The only common reasons to change domain parameters are to increase key size, or purposely introduce on interoperability barrier (e.g. to force an upgrade).

One advantage of fixed domain parameters $(p,q,g)$ in DSA is that it simplifies the choice of a user's key: basically one selects a random private $x\in[1, q–1]$ and computes the public $y=g^x\bmod p$.

As pointed in this other answer, sharing domain parameters increase odds that several users share the same key. However, for a good RNG, $2^n$ users and the lowest security level in FIPS 186, odds that it happens are less then $2^{160-2⋅n}$; that's negligible even for one key per day per human on our planet during a century. And if the RNG is flawed, all bets are off anyway.