ag.algebraic geometry - Are schemes that "have enough locally frees" necessarily separated

Let me motivate my question a bit.

Thm. Let $X$ be a locally noetherian finite-dimensional regular scheme. If $X$ has enough locally frees, then the natural homomorphism $K^0(X)longrightarrow K_0(X)$ is an isomorphism.

A locally noetherian scheme has enough locally frees if every coherent sheaf is the quotient of a locally free coherent sheaf, $K^0(X)$ denotes the Grothendieck group of vector bundles on $X$ and $K_0(X)$ denotes the Grothendieck group of coherent sheaves on $X$.

The above theorem is shown as follows.

By the regularity (and finite-dimensionality!) of $X$, we can construct a finite resolution by a standard procedure. (Surject onto the kernel at each stage with a vector bundle.) Then the "Euler characteristic" associated to this resolution is inverse to the natural morphism.

Now, I was looking through the literature (Weibel's book basically) and I saw that this theorem appears with the additional condition of separability. (Edit: This is not necessary. The point is that noetherian schemes that have enough locally frees are semi-separated.)

Example. Take the projective plane $X$ with a double origin. Then $K^0(X) cong mathbf{Z}^3$ whereas $K_0(X) cong mathbf{Z}^4$.

Example. Take the affine plane $X$ with a double origin. Then $K^0(X) cong mathbf{Z}$, whereas $K_0(X) cong mathbf{Z}oplus mathbf{Z}$.

So I figured I must be missing something...Thus, I ask:

Q. Are locally noetherian schemes that have enough locally frees separated?

EDIT.

The answer to the above question is "No" as the example by Antoine Chambert-Loir shows.

From Philipp Gross's answer, we conclude that a noetherian scheme which has enough locally frees is semi-separated. This means that, for every pair of affine open subsets $U,Vsubset X$, it holds that $Ucap V$ is affine. Note that separated schemes are semi-separated and that Antoine's example is also semi-separated.

Taking a look at Totaro's article cited by Philipp Gross, we see that a regular noetherian scheme which is semi-separated has enough locally frees. (Do regular semi-separated and locally noetherian schemes have enough locally frees?)

This was (in a way) also remarked by Hailong Dao. He mentions the result of Kleiman and independently Illuzie. Recently, Brenner and Schroer observed that their proof works also with $X$ noetherian semi-separated locally $mathbf{Q}$-factorial. See page 4 of Totaro's paper. In short, separated is not really needed but semi-separated is.

Thus, we can conclude the following.

Suppose that $X$ is a regular and finite-dimensional scheme.

If $X$ has enough locally frees, then $K^0(X) longrightarrow K_0(X)$ is an isomorphism. For example, $X$ is noetherian and semi-separated.

Anyway, thanks to everybody for their answers. They helped me alot!