Let $P$ be a regular 2006-gon. We are tasked with finding the maximum number of isosceles triangles that can be formed by dissecting $P$ using 2003 diagonals such that each triangle has two good sides, where a side is called good if it divides the boundary of $P$ into two parts, each having an odd number of sides. The sides of $P$ are also considered to be good.

### Step-by-Step Process:

1. Understanding the Configuration and Properties:
- A regular 2006-gon, $P$, can be divided into non-overlapping triangles using 2003 diagonals. No two of these diagonals should intersect inside the polygon.
- In total, a 2006-gon can be divided into $2006 - 2 = 2004$ triangles.
- We need to focus on forming isosceles triangles with two good sides.

2. Characterizing Good Diagonals:
- A diagonal of $P$ is good if its endpoints divide the polygon into two parts such that each part has an odd number of sides.
- The length of these diagonal-segments must be odd because dividing an even-died polygon into sections with an odd count on either side requires cutting through an odd number of vertices.

3. Counting Good Diagonals:
- To count the number of such diagonals, note that a diagonal connecting vertex $v_i$ to $v_{i+k}$ (where $k \leq 2005$) forms two polygon arcs with lengths $k$ and $2006 - k$.
- Both $k$ and $2006 - k$ must be odd.
- Therefore, $k$ is an odd number less than 2006.
- The odd numbers $k$ range from 1 to 2005, inclusive. There are:
$$\frac{2005 - 1}{2} + 1 = 1003$$
odd numbers.

4. Maximizing Isosceles Triangles:
- We need to ensure that each triangle has two such good sides. Since a triangle is determined by three vertices, and two of its sides need to be good (i.e., our previously defined good diagonals or sides), each triangle can potentially have exactly 2 good sides.

5. Solution Conclusion:
- The maximum number of isosceles triangles, each with two good sides, is related directly to determining the configuration of these 1003 potential good diagonals.
- As diagonals are added one by one across the entire configuration to triangulate the polygon, each new diagonal can create an isosceles triangle with parts of previous triangles.
- Hence, the maximum number of isosceles triangles is:
$$\boxed{1003}$$

This analysis ensures that the maximum number of isosceles triangles that could appear in the given configuration is indeed 1003, conforming to specified conditions of polygon dissection and diagonal configuration.