Let's start the counting today

Numbers encountered by Ashish $\implies$ $3,6,9,12, \dots, 999$

Total numbers encountered= $333$

Numbers encountered by me $\implies$ $5, 10, 15, \dots, 1000$

Total numbers encountered= $200$

Common numbers encountered by both of us will be the multiples of $15$ as we have $3×5=15$

Total multiples of $15 \space till 1000$ $\implies$ $15, 30, 45,..., 990 = 66$

As we both will count these 66 numbers they have to be removed once.

So, Total numbers encountered by both of us provided that number is counted once $= 333+200-66 = 467$

Using A.P.

Ashish counted multiples of 3.

$\Rightarrow 3,6,9,...,999$

Total numbers counted by Ashish is:

$999=3+(n_{1}-1)3$

$n_{1}=333$

I (Abhay) counted multiples of 5.

$\Rightarrow 5,10,15,...1000$

Total numbers counted by me:

$1000=5+(n_{2}-1)5$

$n_{2}=200$

Now, Identical numbers counted by me and Ashish.

These are multiple of 15.

$\Rightarrow 15,30,45,...990$

$990=15+(n_{3}-1)15$

$n_{3}=66$

So, Total numbers counted by me and Ashish are $200+333-66=\boxed{467}$ .