The first blast took place on Sunday afternoon at the Souvantxay market in Khanthabouly town, Savannakhet province, killing two people and injuring at least five others, Radio Free Asia (RFA) said on Wednesday, citing local sources.
The market place was not busy at the time, or else the toll could have been much higher, it added.
The Lao foreign ministry said it had no information on the attack.
The second blast took place late in the evening of the same day, near Vientiane's Khoua Din evening market, after a grenade was hurled by two men on a motorbike, the US government-funded radio network said.
Tuesday's edition of the state-run Vientiane Times acknowledged the incident had taken place, but blamed it on a personal dispute, saying "two gangs were arguing with each other when one gang member took the grenade and exploded it".
The English-language newspaper said no one was hurt in the blast and police were hunting those responsible.
Laos has been rocked by a series of attacks this year which, observers say, could have been carried out by small bands of ethnic minority rebels opposed to the communist regime.
"Two gangs were arguing with each other when one gang member took the grenade and exploded it"
The guerrillas, mainly from the Hmong minority, have waged an ineffective, low-level insurgency for nearly three decades since being abandoned by their US sponsors at the end of the Vietnam War.
The Lao military has been accused by human rights groups of using heavy-handed and often brutal tactics to eliminate them.
Sunday's grenade attack was the second blast to have occurred in the capital in the past couple of months. At least 10 people were injured in an explosion on 4 August at a bus station near the scene of Sunday's blast.
The government has said that the August attack is not politically motivated, in line with its insistence that Laos does not have an insurgency problem.
Many diplomats, however, have cast doubts on the government's claims, pointing to a number of deadly highway ambushes and other incidents in recent months.
Vientiane was gripped by a series of 14 bombings between 2000 and 2001, in which four people were killed and more than 40 injured.
No one claimed responsibility for those attacks, but the the government said it believed they were planned from outside Laos.