The rivers are patroled by
1. the Iraqi DBE Coastal Border Guard,
2. the IA divisional engineer Boat Companies,
3. the IP River Police,
4. and the US Army, Marine, and Navy Boat units.
Put that together with air and the rivers are well covered.
Plenty of arms have been found on islands in the Tigris and Euphraties since 2003. The last report of such a raid was in 2008.
Given the extensive road network in Iraq, the rivers are not the primary transport route compared to Vietnam...
One other detail.
The primary sources of smuggling arms into Iraq are via/from Iran and Syria. The rivers do not support traffic from those sources...