I solved the problem using trigger for allowing user to edit rows in grid as follows:
<Controls:DataGridTemplateColumn Header="Note Title" CanUserSort="True" ToolTipService.ToolTip="Note Title" MaxWidth="250"> <Controls:DataGridTemplateColumn.CellStyle>
<Style TargetType="Controls:DataGridCell">
<Style.Triggers>
<Trigger Property="IsSelected" Value="true">
<Setter Property="Template">
<Setter.Value>
<ControlTemplate>
<TextBox TextWrapping="Wrap">
<TextBox.Text>
<Binding Path="ColumnValue" Mode="TwoWay" />
</TextBox.Text>
</TextBox>
</ControlTemplate>
</Setter.Value>
</Setter>
</Trigger>
<Trigger Property="IsSelected" Value="false">
<Setter Property="Template">
<Setter.Value>
<ControlTemplate>
<TextBlock TextWrapping="Wrap">
<TextBlock.Text>
<Binding Path="ColumnValue" Mode="TwoWay" />
</TextBlock.Text>
</TextBlock>
</ControlTemplate>
</Setter.Value>
</Setter>
</Trigger>
</Style.Triggers>
</Style>
</Controls:DataGridTemplateColumn.CellStyle>
</Controls:DataGridTemplateColumn>
There is also CellEditingTemplate which can be used, but It was not working for me properly. So I used triggers. When user selects row, I set TextBox in the template of column instead if TextBlock.